# Go Math Grade 8 Answer Key Chapter 8 Solving Systems of Linear Equations

Students of Grade 8 can get a detailed explanation for all the questions in Go Math Answer Key Chapter 8 Solving Systems of Linear Equations. In addition to the exercise problems we also provide the solutions for the review test. So, go through all the answers and explanations provided by the math experts in Go Math Grade 8 Chapter 8 Solving Systems of Linear Equations Answer Key. Our aim is to provide easy and simple tricks to solve the problems in Go Math Grade 8 Answer Key Chapter 8 Solving Systems of Linear Equations.

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Lesson 1: Solving Systems of Linear Equations by Graphing

Lesson 2: Solving Systems by Substitution

Lesson 3: Solving Systems by Elimination

Lesson 4: Solving Systems by Elimination with Multiplication

Lesson 5: Solving Solving Special Systems

Model Quiz

Review

### Guided Practice – Solving Systems of Linear Equations by Graphing – Page No. 232

Solve each system by graphing.

Question 1.
$$\left\{\begin{array}{l}y=3 x-4 \\y=x+2\end{array}\right.$$

Type below:
______________

Explanation:
y = 3x – 4
y = x + 2
The solution of the linear system of equations is the intersection point of the two equations.
(3, 5) is the solution of the system of equations.
If x = 3, y = 3(3) – 4 = 9 – 4 = 5; y = 3 + 2 = 5
5 = 5; True

Question 2.
$$\left\{\begin{array}{l}x-3 y=2 \\-3x+9y=-6\end{array}\right.$$

Type below:
______________

Infinitely many solutions

Question 3.
Mrs. Morales wrote a test with 15 questions covering spelling and vocabulary. Spelling questions (x) are worth 5 points and vocabulary questions (y) are worth 10 points. The maximum number of points possible on the test is 100.
a. Write an equation in slope-intercept form to represent the number of questions on the test.

Type below:
______________

y = -x + 15

Explanation:
Mrs. Morales wrote a test with 15 questions covering spelling and vocabulary. Spelling questions (x) are worth 5 points and vocabulary questions (y) are worth 10 points.
x + y = 15
x + y – x = -x + 15
y = -x + 15

Question 3.
b. Write an equation in slope-intercept form to represent the total number of points on the test.
Type below:
______________

y = -1/2 . x + 10

Explanation:
The total number of points on the test is 100
5x + 10y = 100
5x + 10y – 5x = -5x + 100
10y = -5x + 100
y = -5/10 . x + 100/10
y = -1/2 . x + 10

Question 3.
c. Graph the solutions of both equations.
Type below:
______________

Question 3.
d. Use your graph to tell how many of each question type are on the test.
_________ spelling questions
_________ vocabulary questions

10 spelling questions
5 vocabulary questions

ESSENTIAL QUESTION CHECK-IN

Question 4.
When you graph a system of linear equations, why does the intersection of the two lines represent the solution of the system?
Type below:
______________

Solving a system of linear equations means finding the solutions that satisfy all the equations of that system. When we graph a system of linear equations, the intersection point lies on the line of each equation, which means that satisfies all the equations. Therefore, it is considered to be the solution to that system.

### Solving Systems of Linear Equations by Graphing – Page No. 233

Question 5.
Vocabulary
A_________________ is a set of equations that have the same variables.
______________

system of equations

Explanation:
A system of equations is a set of equations that have the same variables.

Question 6.
Eight friends started a business. They will wear either a baseball cap or a shirt imprinted with their logo while working. They want to spend exactly $36 on the shirts and caps. The shirts cost$6 each and the caps cost $3 each. a. Write a system of equations to describe the situation. Let x represent the number of shirts and let y represent the number of caps. ______________ Answer: 6x + 3y = 36 Explanation: The sum of caps and shirts is 8. The total cost of caps and shirts is$36.
x + y = 8
6x + 3y = 36

Question 6.
b. Graph the system. What is the solution and what does it represent?

Type below:
______________

The solution is (4, 4)

Explanation:
x + y – x = -x + 8
y = -x + 8
6x + 3y – 6x = -6x + 36
3y = -6x + 36
y = -6/2 . x + 36/3
y = -2x + 12
(4, 4). They should order 4 shirts and 4 caps.

Question 7.
Multistep The table shows the cost for bowling at two bowling alleys.

a. Write a system of equations, with one equation describing the cost to bowl at Bowl-o-Rama and the other describing the cost to bowl at Bowling Pinz. For each equation, let x represent the number of games played and let y represent the total cost.
Type below:
______________

y = 2.5x + 2
y = 2x + 4

Explanation:
Cost at Bowl-o-Rama => y = 2.5x + 2
Cost at Bowling Pinz => y = 2x + 4

Question 7.
b. Graph the system. What is the solution and what does it represent?

Type below:
______________

Explanation:
The solution of the linear system of equations is the intersection of the two equations.
(4, 12)
When 4 games are played, the total cost is $12. ### Solving Systems of Linear Equations by Graphing – Page No. 234 Question 8. Multi-Step Jeremy runs 7 miles per week and increases his distance by 1 mile each week. Tony runs 3 miles per week and increases his distance by 2 miles each week. In how many weeks will Jeremy and Tony be running the same distance? What will that distance be? Type below: ______________ Answer: After 4 weeks Jeremy and Tony will be running the same distance and that distance would be 11 miles. Explanation: Multi-Step Jeremy runs 7 miles per week and increases his distance by 1 mile each week. y = x + 7 Tony runs 3 miles per week and increases his distance by 2 miles each week. y = 2x + 3 The solution of the system of linear equation is (4, 11) which means that after 4 weeks Jeremy and Tony will be running the same distance and that distance would be 11 miles. Question 9. Critical Thinking Write a real-world situation that could be represented by the system of equations shown below. $$\left\{\begin{array}{l}y=4 x+10 \\y=3x+15\end{array}\right.$$ Type below: ______________ Answer: The entry fee for the first gym is$10 and for every hour that you spend there, you pay an extra $4. If we denote with x the number of hours that somebody spends at the gym and with y the total cost is y = 4x + 10 The entry fee for the second gym is$15 and for every hour that you spend there, you pay an extra $3. If we denote with x the number of hours that somebody spends at the gym and with y the total cost is y = 3x + 15 y = 4x + 10 y = 3x + 15 FOCUS ON HIGHER-ORDER THINKING Question 10. Multistep The table shows two options provided by a high-speed Internet provider. a. In how many months will the total cost of both options be the same? What will that cost be? ________ months$ ________

5 months
$200 Explanation: Let y be the total cost after x month y = 30x + 50 Let y be the total cost after x month y = 40x Substitute y = 40x in y = 30x + 50 40x = 30x + 50 40x – 30x = 50 10x = 50 x = 50/10 x = 5 The total cost of both options will be the same after 5 months. Total cost would be y = 40(5) =$200.

Question 10.
b. If you plan to cancel your Internet service after 9 months, which is the cheaper option? Explain.
______________

When x = 9 months
y = 30(9) + 50 = $320 y = 40(9) =$360
$320 <$360
Option 1 is cheaper as the total cost is less for option 1

### Guided Practice – Solving Systems by Substitution – Page No. 240

Solve each system of linear equations by substitution.

Question 1.
$$\left\{\begin{array}{l}3x-2y=9 \\y=2x-7\end{array}\right.$$
x = ________
y = ________

x = 5
y = 3

Explanation:
$$\left\{\begin{array}{l}3x-2y=9 \\y=2x-7\end{array}\right.$$
Substitute 2x – 7 in 3x – 2y = 9
3x – 2(2x – 7) = 9
3x – 4x + 14 = 9
-x + 14 = 9
-x + 14 – 14 = 9 – 14
-x = -5
x = -5/-1 = 5
y = 2(5) – 7 = 3
The solution is (5, 3)

Solve each system. Estimate the solution first.

Question 5.
$$\left\{\begin{array}{l}6x+y=4 \\x-4y=19\end{array}\right.$$
Estimate ______________
Solution ______________
Type below:
______________

Estimate (2, -5)
Solution (1.4, -4.4)

Explanation:
$$\left\{\begin{array}{l}6x+y=4 \\x-4y=19\end{array}\right.$$
Let’s find the estimation by graphing the equations
Estimate: (2, -5)

x = 4y + 19
6(4y + 19) + y = 4
24y + 114 + y = 4
25y + 114 = 4
25y = 4 – 114
25y = -110
y = -110/25
y = -4.4
x + 4(-4.4) = 19
x + 17.6 = 19
x = 19 – 17.6
x = 1.4
The solution is (1.4, -4.4)

Question 6.
$$\left\{\begin{array}{l}x+2y=8 \\3x+2y=6\end{array}\right.$$
Estimate ______________
Solution ______________
Type below:
______________

Estimate (-1, 5)
Solution (-1, 4.5)

Explanation:
$$\left\{\begin{array}{l}x+2y=8 \\3x+2y=6\end{array}\right.$$
Let’s find the estimation by graphing the equations
Estimate: (-1, 5)

x = -2y + 8
Substitute the equation x = -2y + 8 in 3x + 2y = 6
3(-2y + 8) + 2y = 6
-6y + 24 + 2y = 6
-4y = 6 – 24
-4y = -18
y = -18/-4
y = 4.5
x + 2(4.5) = 8
x + 9 = 8
x = 8 – 9
x = -1
The solution is (-1, 4.5)

Question 7.
$$\left\{\begin{array}{l}3x+y=4 \\5x-y=22\end{array}\right.$$
Estimate ______________
Solution ______________
Type below:
______________

Estimate (3, -6)
Solution (3.25, -5.75)

Explanation:
$$\left\{\begin{array}{l}3x+y=4 \\5x-y=22\end{array}\right.$$
Find the Estimation using graphing the equations.
Estimate: (3, -6)

y = -3x + 4
Substitute y = -3x + 4 in 5x – y = 22
5x – (-3x + 4) = 22
5x + 3x -4 = 22
8x = 26
x = 26/8
x = 3.25
3(3.25) + y = 4
9.75 + y = 4
y = 4 – 9.75
y = -5.75
The solution is (3.25, -5.75)

Question 8.
$$\left\{\begin{array}{l}2x+7y=2 \\x+y=-1\end{array}\right.$$
Estimate ______________
Solution ______________
Type below:
______________

Estimate (-2, 1)
Solution (-1.8, 0.8)

Explanation:
$$\left\{\begin{array}{l}2x+7y=2 \\x+y=-1\end{array}\right.$$
Find the Estimation using graphing the equations.
Estimate: (-2, 1)

y = -x -1
Substitute y = -x – 1 in 2x + 7y = 2
2x + 7(-x – 1) = 2
2x – 7x -7 = 2
-5x = 2 + 7
-5x = 9
x = -9/5
x = -1.8
-1.8 + y = -1
y = -1 + 1.8
y = 0.8
The solution is (-1.8, 0.8)

Question 9.
Adult tickets to Space City amusement park cost x dollars. Children’s tickets cost y dollars. The Henson family bought 3 adult and 1 child tickets for $163. The Garcia family bought 2 adult and 3 child tickets for$174.
a. Write equations to represent the Hensons’ cost and the Garcias’ cost.
Hensons’ cost: ________________
Garcias’ cost:__________________
Type below:
______________

Hensons’ cost: 3x + y = 163
Garcias’ cost: 2x + 3y = 174

Explanation:
Henson’s cost
3x + y = 163
Garcia’s cost
2x + 3y = 174

Question 9.
b. Solve the system.
adult ticket price: $_________ Garcias’ cost:$ _________

adult ticket price: $45 Garcias’ cost:$ 28

Explanation:
y = -3x + 163
Substitute y = -3x + 163 in 2x + 3y = 174
2x + 3(-3x + 163) = 174
2x -9x + 489 = 174
-7x = -315
x = -315/-7 = 45
3(45) + y = 163
135 + y = 163
y = 163 – 135
y = 28
adult ticket price: $45 Garcias’ cost:$ 28

ESSENTIAL QUESTION CHECK-IN

### 8.2 Independent Practice – Solving Systems by Substitution – Page No. 241

Question 11.
Check for Reasonableness Zach solves the system
$$\left\{\begin{array}{l}x+y=-3 \\x-y=1\end{array}\right.$$
and finds the solution (1, -2). Use a graph to explain whether Zach’s solution is reasonable.

Type below:
______________

Explanation:
$$\left\{\begin{array}{l}x+y=-3 \\x-y=1\end{array}\right.$$
The x coordinate of the solution is negative, hence Zach’s solution is not reasonable.

Represent Real-World Problems Angelo bought apples and bananas at the fruit stand. He bought 20 pieces of fruit and spent $11.50. Apples cost$0.50 and bananas cost $0.75 each. a. Write a system of equations to model the problem. (Hint: One equation will represent the number of pieces of fruit. A second equation will represent the money spent on the fruit.) Type below: ______________ Answer: x + y = 20 0.5x + 0.75y = 11.5 Explanation: x + y = 20 0.5x + 0.75y = 11.5 where c is the number of Apples and y is the number of Bananas. Question 12. b. Solve the system algebraically. Tell me how many apples and bananas Angelo bought. ________ apples ________ bananas Answer: 14 apples 6 bananas Explanation: y = -x + 20 Substitute y = -x + 20 in 0.5x + 0.75y = 11.5 0.5x + 0.75(-x + 20) = 11.5 0.5x – 0.75x + 15 = 11.5 -0.25x + 15 = 11.5 -0.25x = 11.5 – 15 -0.25x = -3.5 x = -3.5/-0.25 x = 14 14 + y = 20 y = 6 Angelo bought 14 apples and 6 bananas. Question 14. Multistep The graph shows a triangle formed by the x-axis, the line 3x−2y=0, and the line x+2y=10. Follow these steps to find the area of the triangle. a. Find the coordinates of point A by solving the system $$\left\{\begin{array}{l}3x-2y=0 \\x-2y=10\end{array}\right.$$ Point A: ____________________ Type below: ______________ Answer: Point A: (2.5, 3.75)Coordinate of A is (2.5, 3.75) Explanation: $$\left\{\begin{array}{l}3x-2y=0 \\x-2y=10\end{array}\right.$$ x = -2y + 10 Substitute x = -2y + 10 in 3x – 2y = 0 3(-2y + 10) -2y = 0 -6y + 30 – 2y = 0 -8y = -30 y = -30/-8 = 3.75 x + 2(3.75) = 10 x + 7.5 = 10 x = 10 – 7.5 x = 2.5 Coordinate of A is (2.5, 3.75) Question 14. b. Use the coordinates of point A to find the height of the triangle. height:__________________ height: $$\frac{□}{□}$$ units Answer: height: 3.75 height: $$\frac{15}{4}$$ units Explanation: The height of the triangle is the y coordinate of A Height = 3.75 Question 14. c. What is the length of the base of the triangle? base:________________ base: ______ units Answer: base: 10 units Explanation: Length of the base = 10 Question 14. d. What is the area of the triangle? A = ______ $$\frac{□}{□}$$ square units Answer: A = 18.75 square units A = 18 $$\frac{3}{4}$$ square units Explanation: Area of the triangle = 1/2 . Height . Base Area = 1/2 . 3.75 . 10 = 18.75 ### Solving Systems by Substitution – Page No. 242 Question 15. Jed is graphing the design for a kite on a coordinate grid. The four vertices of the kite are at A(−$$\frac{4}{3}$$, $$\frac{2}{3}$$), B($$\frac{14}{3}$$, −$$\frac{4}{3}$$), C($$\frac{14}{3}$$, −$$\frac{16}{3}$$), and D($$\frac{2}{3}$$, −$$\frac{16}{3}$$). One kite strut will connect points A and C. The other will connect points B and D. Find the point where the struts cross. Type below: ______________ Answer: The struts cross as (8/3, 10/3) FOCUS ON HIGHER ORDER THINKING Question 17. Communicate Mathematical Ideas Explain the advantages, if any, that solving a system of linear equations by substitution has over solving the same system by graphing. Type below: ______________ Answer: The advantage of solving a system of linear equations by graphing is that it is relatively easy to do and requires very little algebra. Question 18. Persevere in Problem-Solving Create a system of equations of the form $$\left\{\begin{array}{l}Ax+By=C \\Dx+Ey=F\end{array}\right.$$ that has (7, −2) as its solution. Explain how you found the system. Type below: ______________ Answer: x + y = 5 x – y = 9 solves in : x = (5+9)/2 = 7 y = 5-9)/2 = -2 A=1, B=2, C= 5 D=1, E= -1, F=9 x = 7 y = -2 IS a system (even if it is a trivial one) of equations so this answer would be acceptable. The target for a system is to find its SOLUTION SET and not to conclude with x=a and y=b ### Guided Practice – Solving Systems by Elimination – Page No. 248 Question 1. Solve the system $$\left\{\begin{array}{l}4x+3y=1 \\x-3y=-11\end{array}\right.$$ by adding. Type below: ______________ Answer: 4x + 3y = 1 x – 3y = -11 Add the above two equations 4x + 3y = 1 +(x – 3y = -11) Add to eliminate the variable y 5x + 0y = -10 Simplify and solve for x 5x = -10 Divide both sides by 5 x = -10/5 = -2 Substitute into one of the original equations and solve for y. 4(-2) + 3y = 1 -8 + 3y = 1 3y = 9 y = 9/3 = 3 So, (-2, 3) is the solution of the system. Solve each system of equations by adding or subtracting. Question 4. $$\left\{\begin{array}{l}-4x-5y=7 \\3x+5y=-14\end{array}\right.$$ (________ , ________) Answer: (7, -7) Explanation: $$\left\{\begin{array}{l}-4x-5y=7 \\3x+5y=-14\end{array}\right.$$ Add the equations -4x – 5y = 7 +(3x + 5y = -14) y is eliminated as it has reversed coefficients. Solve for x -4x -5y +3x + 5y = 7 -14 -x = -7 x = -7/-1 = 7 Substituting x in either of the equation to find y 3(7) + 5y = -14 21 + 5y -21 = -14 -21 5y = -35 y = -35/5 = -7 The answer is (7, -7) Question 5. $$\left\{\begin{array}{l}x-2y=-19 \\5x+2y=1\end{array}\right.$$ (________ , ________) Answer: (-3, 8) Explanation: $$\left\{\begin{array}{l}x-2y=-19 \\5x+2y=1\end{array}\right.$$ Add the equations x – 2y = -19 +(5x + 2y = 1) y is eliminated as it has reversed coefficients. Solve for x x – 2y + 5x + 2y = -19 + 1 6x = -18 x = -18/6 = -3 Substituting x in either of the equation to find y -3 -2y = -19 -3 -2y + 3 = -19 + 3 -2y = -16 y = -16/-2 = 8 The answer is (-3, 8) Question 8. The Green River Freeway has a minimum and a maximum speed limit. Tony drove for 2 hours at the minimum speed limit and 3.5 hours at the maximum limit, a distance of 355 miles. Rae drove 2 hours at the minimum speed limit and 3 hours at the maximum limit, a distance of 320 miles. What are the two-speed limits? a. Write equations to represent Tony’s distance and Rae’s distance. Type below: ______________ Answer: Tony’s distance: 2x + 3.5y = 355 Rae’s distance: 2x + 3y = 320 where x is the minimum speed and y is the maximum speed. Question 8. b. Solve the system. minimum speed limit:______________ maximum speed limit______________ minimum speed limit: ________ mi/h maximum speed limit: ________ mi/h Answer: minimum speed limit:55 maximum speed limit70 minimum speed limit: 55mi/h maximum speed limit: 70mi/h Explanation: Subtract the equations 2x + 3.5y = 355 -(2x + 3y = 320) x is eliminated as it has reversed coefficients. Solve for y 2x + 3.5y – 2x – 3y = 355 – 320 0.5y = 35 y = 35/0.5 = 70 Substituting y in either of the equation to find x 2x + 3(70) = 320 2x + 210 – 210 = 320 – 210 2x = 110 x = 110/2 = 55 Minimum speed limit: 55 miles per hour Maximum speed limit: 70 miles per hour ESSENTIAL QUESTION CHECK-IN Question 9. Can you use addition or subtraction to solve any system? Explain. ________ Answer: No. One of the variables should have the same coefficient in order to add or subtract the system. ### 8.3 Independent Practice – Solving Systems by Elimination – Page No. 249 Question 10. Represent Real-World Problems Marta bought new fish for her home aquarium. She bought 3 guppies and 2 platies for a total of$13.95. Hank also bought guppies and platies for his aquarium. He bought 3 guppies and 4 platies for a total of $18.33. Find the price of a guppy and the price of a platy. Guppy:$ ________
Platy: $________ Answer: Guppy:$ 3.19
Platy: $2.19 Explanation: 3x + 2y = 13.95 3x + 4y = 18.33 where x is the unit price of guppy and y is the unit price of platy Subtract the equations 3x + 2y = 13.95 -(3x + 4y = 18.33) x is eliminated as it has reversed coefficients. Solve for y 3x + 2y – 3x – 4y = 13.95 – 18.33 -2y = -4.38 y = -4.38/-2 = 2.19 Substituting y in either of the equation to find x 3x + 2(2.19) = 13.95 3x + 4.38 – 4.38 = 13.95 – 4.38 3x = 9.57 x = 9.57/3 = 3.19 The price of a guppy is$3.19 and the price of platy is $2.19 Question 13. Represent Real-World Problems Two cars got an oil change at the same auto shop. The shop charges customers for each quart of oil plus a flat fee for labor. The oil change for one car required 5 quarts of oil and cost$22.45. The oil change for the other car required 7 quarts of oil and cost $25.45. How much is the labor fee and how much is each quart of oil? Labor fee:$ ________
Quart of oil: $________ Answer: Labor fee:$ 14.95
Quart of oil: $1.5 Explanation: 5x + y = 22.45 7x + y = 25.45 where x is the unit cost of quarts of oil and y is the flat fee for labor Subtract the equations 5x + y = 22.45 -(7x + y = 25.45) y is eliminated as it has reversed coefficients. Solve for x 5x + y – 7x – y = 22.45 – 25.45 -2x = -3 x = -3/-2 = 1.5 Substituting x in either of the equation to find y 5(1.5) + y = 22.45 7.5 + y – 7.5 = 22.45 – 7.5 y = 14.95 The labor fee is$14.95 and the unit cost of a quart of oil is $1.5 Question 14. Represent Real-World Problems A sales manager noticed that the number of units sold for two T-shirt styles, style A and style B, was the same during June and July. In June, total sales were$2779 for the two styles, with A selling for $15.95 per shirt and B selling for$22.95 per shirt. In July, total sales for the two styles were $2385.10, with A selling at the same price and B selling at a discount of 22% off the June price. How many T-shirts of each style were sold in June and July combined? ________ T-shirts of style A and style B were sold in June and July. Answer: 15.95x + 22.95y = 2779 15.95x + 17.9y = 2385.10 where x is the number of style A shirts and y is the number of style B shirts In July, the price of style B shirt is 22% of the price of style B shirt in June, hence 0.78(22.95) = 17.90 Subtract the equations 15.95x + 22.95y = 2779 -(15.95x + 17.9y = 2385.10) x is eliminated as it has reversed coefficients. Solve for y 15.95x + 22.95 – 15.95x – 17.9y = 2779 – 2385.10 5.05y = 393.9 y = 393.9/5.05 = 78 Substituting y in either of the equation to find x 15.95x +22.95(78) = 2779 15.95x + 1790.1 – 1790.1 = 2779 – 1790.1 15.95x = 988.9 x = 988.9/15.95 = 62 The number of styles A T shirts sold in June is 62. Since the number of T-shirts sold in both numbers is the same, the total number = 2. 62 = 124. The number of style B T-shirts sold in June is 78. Since the number of T-shirts sold in both numbers is the same, the total number = 2. 78 = 156. Question 15. Represent Real-World Problems Adult tickets to a basketball game cost$5. Student tickets cost $1. A total of$2,874 was collected on the sale of 1,246 tickets. How many of each type of ticket were sold?
img 14
________ student tickets

839 student tickets

Explanation:
x + y = 1246
5x + y = 2874
where x is the number of adult tickets sold and y is the number of student tickets sold.
Subtract the equations
x + y = 1246
-(5x + y = 2874)
y is eliminated as it has reversed coefficients. Solve for x
x + y – 5x – y = 1246 – 2874
-4x = -1628
x = -1628/-4 = 407
Substituting x in either of the equation to find y
407 + y = 1246
407 + y – 407 = 1246 – 407
y = 839
The number of adult tickets sold is 407 and student tickets sold is 839.

### FOCUS ON HIGHER ORDER THINKING – Solving Systems by Elimination – Page No. 250

Question 17.
Jenny used substitution to solve the system
$$\left\{\begin{array}{l}2x+y=8 \\x-y=1\end{array}\right.$$. Her solution is shown below.
Step 1: y = -2x + 8               Solve the first equation for y.
Step 2: 2x + (-2x + 8) = 8     Substitute the value of y in an original equation.
Step 3: 2x – 2x + 8 = 8          Use the Distributive Property.
Step 4: 8 = 8                         Simplify.
a. Explain the Error Jenny made. Describe how to correct it.
Type below:
______________

2x + y = 8
x – y = 1
The rewritten equation should be substituted in the other original equation
Error is that Jenny solved for y in the first equation and substitute it in the original equation.
x – (-2x + 8) = 1
3x – 8 = 1
3x = 9
x = 9/3 = 3
x = 3

Question 17.
b. Communicate Mathematical Ideas Would adding the equations have been a better method for solving the system? If so, explain why.
________

Yes

Explanation:
As the coefficient, if variable y is the opposite, it will be eliminated and solved for x in less number of steps.

### Guided Practice – Solving Systems by Elimination with Multiplication – Page No. 256

Question 1.
Solve the system
$$\left\{\begin{array}{l}3x-y=8 \\-2x+4y=-12\end{array}\right.$$

Type below:
______________

$$\left\{\begin{array}{l}3x-y=8 \\-2x+4y=-12\end{array}\right.$$
Multiply each term in the first equation by 4 to get opposite coefficients for the y-terms.
4(3x – y = 8)
12x – 4y = 32
Add the second equation to the new equation
12x – 4y = 32
+(-2x + 4y = -12)
Add to eliminate the variable y
10x = 20
Divide both sides by 10
x = 20/10 = 2
Substitute into one of the original equations and solve for y
y = 3(2) – 8 = -1
S0, (2, -2)is the solution of the system.

Solve each system of equations by multiplying first.

Question 4.
$$\left\{\begin{array}{l}2x+8y=21 \\6x-4y=14\end{array}\right.$$
Type below:
______________

The solution is (3.5, 1.75)

Explanation:
$$\left\{\begin{array}{l}2x+8y=21 \\6x-4y=14\end{array}\right.$$
To eliminate y terms, multiply the 2nd equation by 2
2(6x – 4y = 14)
2x + 8y = 21
2x + 8y = 21
+(12x – 8y = 28)
y is eliminated it has reversed coefficients. Solve for x
2x + 8y + 12x – 8y = 21 + 28
14x = 49
x = 49/14 = 3.5
Substituting x in either of the equation to find y
6(3.5) – 4y = 14
21 – 4y – 21 = 14 – 21
-4y = -7
y = -7/-4 = 1.75
The solution is (3.5, 1.75)

Question 5.
$$\left\{\begin{array}{l}2x+y=3 \\-x+3y=-12\end{array}\right.$$
(________ , ________ )

Explanation:
$$\left\{\begin{array}{l}2x+y=3 \\-x+3y=-12\end{array}\right.$$
To eliminate x terms, multiply the 2nd equation by 2
2(-x + 3y = -12)
-2x + 6y = -24
2x + y = 3
+(-2x + 6y = -24)
x is eliminated it has reversed coefficients. Solve for y
2x + y – 2x + 6y = 3 – 24
7y = -21
y = -21/7 = -3
Substituting y in either of the equation to find x
-x + 3(-3) = -12
-x -9 + 9 = -12 + 9
-x = -3
x = 3
The solution is (3, -3)

Question 8.
Bryce spent $5.26 on some apples priced at$0.64 each and some pears priced at $0.45 each. At another store, he could have bought the same number of apples at$0.32 each and the same number of pears at $0.39 each, for a total cost of$3.62. How many apples and how many pears did Bryce buy?
a. Write equations to represent Bryce’s expenditures at each store
First store: _____________
Second store: _____________
Type below:
_____________

First store: 0.64x + 0.45y = 5.26
Second store: 0.32x + 0.39y = 3.62

Explanation:
First store = 0.64x + 0.45y = 5.26
Second store = 0.32x + 0.39y = 3.62
where x is the number of apples and y is the number of pears.

Question 8.
b. Solve the system.
Number of apples: _______
Number of pears: _______

Number of apples: 4
Number of pears: 6

Explanation:
First store = 0.64x + 0.45y = 5.26
Second store = 0.32x + 0.39y = 3.62
Multiply by 100
64x + 45y = 526
32x + 39y = 362
To eliminate x terms, multiply the 2nd equation by 2
2(32x + 39y = 362)
64x + 45y = 526
Subtract the equations
64x + 45y = 526
-(64x + 78y = 724)
x is eliminated it has reversed coefficients. Solve for y
64x + 45y – 64x – 78y = 526 – 724
-33y = -198
y = -198/-33 = 6
Substituting y in either of the equation to find x
32x + 39(6) = 362
32x + 234 – 234 = 362 – 234
32x = 128
x = 128/32 = 4
He bought 4 apples and 6 pears.

ESSENTIAL QUESTION CHECK-IN

### Solving Systems by Elimination with Multiplication – Page No. 257

Question 10.
Explain the Error Gwen used elimination with multiplication to solve the system
$$\left\{\begin{array}{l}2x+6y=3 \\x-3y=-1\end{array}\right.$$
Her work to find x is shown. Explain her error. Then solve the system.
2(x − 3y) = -1
2x − 6y = -1
+2x + 6y = 3
_____________
4x + 0y = 2
x = $$\frac{1}{2}$$
Type below:
____________

2x + 6y = 3
x – 3y = -1
To eliminate x terms, multiply the 2nd equation by 2
2(x – 3y = -1)
2x – 6y = -2
Error is the Gnew did not multiply the entire expression with 2.
2x + 6y = 3
+(2x – 6y = -2)
y is eliminated it has reversed coefficients. Solve for x
2x + 6y + 2x – 6y = 3 – 2
4x = 1
x = 1/4
Substituting x in either of the equation to find y
x – 3y = -1
1/4 – 3y – 1/4 = -1 -1/4
-3y = -5/4
y = -5/4(-3) = 5/12

Question 11.
Represent Real-World Problems At Raging River Sports, polyester-fill sleeping bags sell for $79. Down-fill sleeping bags sell for$149. In one week the store sold 14 sleeping bags for $1,456. a. Let x represent the number of polyester-fill bags sold and let y represent the number of down-fill bags sold. Write a system of equations you can solve to find the number of each type sold. Type below: ____________ Answer: x + y = 14 79x + 149y = 1456 where x is the polyester-fill bags and y is the number of down-fill bags Question 11. b. Explain how you can solve the system for y by multiplying and subtracting. Type below: ____________ Answer: x + y = 14 79x + 149y = 1456 Multiply the second equation by 79. Subtract the new equation from the first equation and solve the resulting equation for y. Question 11. c. Explain how you can solve the system for y using substitution. Type below: ____________ Answer: Solve the second equation for x. Substitute the expression for x , in the first equation and solve the resulting equation for y. Question 11. d. How many of each type of bag were sold? _______ polyester-fill _______ down-fill Answer: 9 polyester-fill 5 down-fill Explanation: x + y = 14 79x + 149y = 1456 To eliminate x terms, multiply the 2nd equation by 2 79(x + y = 14) 79x + 149y = 1456 Subtract the equations 79x + 79y = 1106 -(79x + 149y = 1456) x is eliminated it has reversed coefficients. Solve for y 79x + 79y – 79x – 149y = 1106 – 1456 -70y = -350 y = -350/-70 = 5 Substituting y in either of the equation to find x x + 5 = 14 x = 14 – 5 x = 9 There were 9 polyester-fill bags and 5 down-fill bags sold. ### Solving Systems by Elimination with Multiplication – Page No. 258 Question 13. Represent Real-World Problems A farm stand sells apple pies and jars of applesauce. The table shows the number of apples needed to make a pie and a jar of applesauce. Yesterday, the farm picked 169 Granny Smith apples and 95 Red Delicious apples. How many pies and jars of applesauce can the farm make if every apple is used? _______ pies _______ jars of applesauce Answer: 21 pies 16 jars of applesauce Explanation: 5x + 4y = 169 3x + 2y = 95 where x is the number of apples needed for pie and y is the number of apples for jar of applesauce To eliminate y terms, multiply the 2nd equation by 2 2(3x + 2y = 95) 6x + 4y = 190 Subtract the equations 5x + 4y = 169 – (6x + 4y = 190) y is eliminated it has reversed coefficients. Solve for x 5x + 4y – 6x – 4y = 169 – 190 -x = -21 x = -21/-1 = 21 Substituting x in either of the equation to find y 5(21) + 4y = 169 105 + 4y – 105 = 169 – 105 4y = 64 y = 64/4 = 16 The number of apples needed for pie is 21 and the number of apples for a jar of applesauce is 16. FOCUS ON HIGHER-ORDER THINKING Question 15. Consider the system $$\left\{\begin{array}{l}2x+3y=6 \\3x+7y=-1\end{array}\right.$$ a. Communicate Mathematical Ideas Describe how to solve the system by multiplying the first equation by a constant and subtracting. Why would this method be less than ideal? Type below: ____________ Answer: Multiplying the first equation by a constant and subtracting 2x + 3y = 6 3x + 7y = -1 Multiply the first equation by 1.5 and subtract. This would be less than ideal because you would introduce decimals into the solution process. Question 15. b. Draw Conclusions Is it possible to solve the system by multiplying both equations by integer constants? If so, explain how. Type below: ____________ Answer: Yes Explanation: Multiply the first equation by 3 and the second equation by 2. Both x-term coefficients would be 6. Solve by eliminating the x-terms using subtraction. Question 15. c. Use your answer from part b to solve the system. (_______ , _______) Answer: (9, -4) Explanation: 2x + 3y = 6 3x + 7y = -1 Multiply the first equation by 3 and the second equation by 2. 3(2x + 3y = 6) 2(3x + 7y = -1) Subtract the equations 6x + 9y = 18 -(6x + 14y = -2) x is eliminated it has reversed coefficients. Solve for y 6x + 9y – 6x – 14y = 18 + 2 -5y = 20 y = 20/-5 = -4 Substituting y in either of the equation to find x 2x + 3(-4) = 6 2x = 18 x = 18/2 = 9 The solution is (9, -4) ### Guided Practice – Solving Solving Special Systems – Page No. 262 Use the graph to solve each system of linear equations Question 1. A. $$\left\{\begin{array}{l}4x-2y=-6 \\2x-y=4\end{array}\right.$$ B. $$\left\{\begin{array}{l}4x-2y=-6 \\x+y=6\end{array}\right.$$ C. $$\left\{\begin{array}{l}2x-y=4 \\6x-3y=-12\end{array}\right.$$ STEP 1 Decide if the graphs of the equations in each system intersect, are parallel, or are the same line. System A: The graphs __________ System B: The graphs __________ System C: The graphs __________ Answer: System A: The graphs are parallel System B: The graphs are intersecting System C: The graphs are the same line Explanation: System A: 4x – 2y = -6 2x – y = 4 System B: 4x – 2y = -6 x + y = 6 System C: 2x – y = 4 6x – 3y = 12 Question 1. STEP 2 Decide how many points the graphs have in common. a. Intersecting lines have _______________ point(s) in common. b. Parallel lines have _______________ point(s) in common. c. The same lines have ___________ point(s) in common. a. __________ b. __________ c. __________ Answer: a. Intersecting lines have one point(s) in common. b. Parallel lines have no point(s) in common. c. The same lines have infinitely many points (s) in common. Explanation: From the graphs, Intersecting lines have one point(s) in common Parallel lines have no point(s) in common The same lines have infinitely many points (s) in common Question 1. STEP 3 Solve each system. System A has __________ points in common, so it has __________ solution. System B has __________ points in common. That point is the solution, __________. System C has __________ points in common. ________ ordered pairs on the line will make both equations true. Type below: ___________ Answer: System A has no points in common, so it has no solution. System B has one point in common. That point is the solution, (1,5). System C has an infinite number of points in common. All ordered pairs on the line will make both equations true. Explanation: Number of solutions for each system System A has no points in common, so it has no solution. System B has one point in common. That point is the solution, (1,5). System C has an infinite number of points in common. All ordered pairs on the line will make both equations true. Solve each system. Tell how many solutions each system has. Question 4. $$\left\{\begin{array}{l}6x-2y=-10 \\3x+4y=-25\end{array}\right.$$ ___________ Answer: one solution Explanation: 6x – 2y = -10 3x + 4y = -25 To eliminate y terms, multiply the 1st equation by 2 2(6x – 2y = -10) 12x – 4y = -20 Add the equations 12x – 4y = -20 +(3x + 4y = -25) y is eliminated as it has reversed coefficients. Solve for x. 12x – 4y + 3x + 4y = -20 – 25 15x = -45 x = -45/15 = -3 Substitute x in any one of the original equations and solve for y 3(-3) + 4y = -25 -9 + 4y + 9 = -25 + 9 4y = -16 y = -16/4 y = -4 There is one solution, (-3, -4) ESSENTIAL QUESTION CHECK-IN Question 5. When you solve a system of equations algebraically, how can you tell whether the system has zero, one, or an infinite number of solutions? Type below: ___________ Answer: When x and y are eliminated and the statement is true, the system has infinitely many solutions. When x and y are eliminated and the statement is false, the system has no solutions. When the system has one solution by solving, the system has one solution. ### 8.5 Independent Practice – Solving Solving Special Systems – Page No. 263 Solve each system by graphing. Check your answer algebraically. Question 6. $$\left\{\begin{array}{l}-2x+6y=12 \\x-3y=3\end{array}\right.$$ Solution: ______________ ___________ Answer: $$\left\{\begin{array}{l}-2x+6y=12 \\x-3y=3\end{array}\right.$$ Graph the equations on same coordinate plane No solution as equations are parallel To eliminate y terms, multiply the 2nd equation by 2 2(x – 3y = 3) 2x – 6y = 6 Add the equations -2x + 6y = 12 2x – 6y = 6 x and y are eliminated as it has reversed coefficients. -2x + 6y + 2x – 6y = 12 + 6 0 = 18 The statement is false, hence the system has no solution. Question 7. $$\left\{\begin{array}{l}15x+5y=5 \\3x+y=1\end{array}\right.$$ Solution: ______________ ___________ Answer: $$\left\{\begin{array}{l}15x+5y=5 \\3x+y=1\end{array}\right.$$ Graph the equations on the same coordinate plane Infinitely many solutions as equations are overlapping To eliminate y terms, multiply the 2nd equation by 5 5(3x + y = 1) 15x + 5y = 5 Subtract the equations 15x + 5y = 5 -(15x + 5y = 5) x and y is eliminated as it has reversed coefficients. 15x + 5y -15x – 5y = 5 – 5 0 = 0 The statement is true, hence the system has infinitely many solutions. For Ex. 8 14, state the number of solutions for each system of linear equations Question 8. a system whose graphs have the same slope but different y-intercepts ___________ Answer: No solutions Explanation: Equations are parallel No solutions Question 12. the system $$\left\{\begin{array}{l}y=2 \\y=-3\end{array}\right.$$ ___________ Answer: No solutions Explanation: Equations are parallel No solutions Question 13. the system $$\left\{\begin{array}{l}y=2 \\y=-3\end{array}\right.$$ ___________ Answer: One solution Explanation: Equations are intersecting One solution Question 14. the system whose graphs were drawn using these tables of values: ___________ Answer: No solutions Explanation: Equations are parallel The slope is the same for both equations but the y-intercept is different. No solutions ### Solving Solving Special Systems – Page No. 264 Question 16. Represent Real-World Problems Two school groups go to a roller skating rink. One group pays$243 for 36 admissions and 21 skate rentals. The other group pays \$81 for 12 admissions and 7 skate rentals. Let x represent the cost of admission and let y represent the cost of a skate rental. Is there enough information to find values for x and y? Explain.

___________

36x + 21y = 243
12x + 7y = 81
where x is the cost of admission and y is the cost of stake rentals.
Although the information can be used to develop a system of linear equations, where each equation has two variables when the system is solved, the number of solutions is infinite, Mee the values of x and y cannot be determined.
No

Question 17.
Represent Real-World Problems Juan and Tory are practicing for a track meet. They start their practice runs at the same point, but Tory starts 1 minute after Juan. Both run at a speed of 704 feet per minute. Does Tory catch up to Juan? Explain.
___________

No; Both Juan and Tory run at the same rate, so the lines representing the distances each has run are parallel. There is no solution to the system

FOCUS ON HIGHER-ORDER THINKING

Question 18.
Justify Reasoning A linear system with no solution consists of the equation y = 4x − 3 and a second equation of the form y = mx + b. What can you say about the values of m and b? Explain your reasoning.
Type below:
___________

y = 4x – 3
y = mx + b
Since the system has no solutions, the two equations are parallel. The value of the slope, m would be the same i.e. 4. The value of y-intercept, b can be any number except -3 as b is different for parallel lines.

Question 19.
Justify Reasoning A linear system with infinitely many solutions consists of equation 3x + 5 = 8 and a second equation of the form Ax + By = C. What can you say about the values of A, B, and C? Explain your reasoning.
Type below:
___________

3x + 5 = 8
Ax + By = C
Since the system has infinitely many solutions, the values of A, B, and C must all be the same multiple of 3, 5, and 8, respectively. The two equations represent a single line, so the coefficients and constants of one equation must be a multiple of the other.

Question 20.
Draw Conclusions Both points (2, -2) and (4, -4) are solutions to a system of linear equations. What conclusions can you make about the equations and their graphs?
Type below:
___________

If a system has more than one solution, the equations represent the same line and have infinitely many solutions.

### Ready to Go On? – Model Quiz – Page No. 265

8.1 Solving Systems of Linear Equations by Graphing

Solve each system by graphing.

Question 1.
$$\left\{\begin{array}{l}y=x-1 \\y=2x-3\end{array}\right.$$

(________ , ________)

(2, 1)

Explanation:
y = x – 1
y = 2x – 3
Graph the equations on the same coordinate plane

The solution of the system is the point of intersection
The solution is (2, 1)

Question 2.
$$\left\{\begin{array}{l}x+2y=1 \\-x+y=2\end{array}\right.$$

(________ , ________)

(-1, 1)

Explanation:
x + 2y = 1
-x + y = 2
Graph the equations on the same coordinate plane

The solution of the system is the point of intersection
The solution is (-1, 1)

8.2 Solving Systems by Substitution

Solve each system of equations by substitution.

8.3 Solving Systems by Elimination

Solve each system of equations by adding or subtracting.

Question 5.
$$\left\{\begin{array}{l}3x+y=9 \\2x+y=5\end{array}\right.$$
(________ , ________)

(4, -3)

Explanation:
$$\left\{\begin{array}{l}3x+y=9 \\2x+y=5\end{array}\right.$$
Subtract the equations
3x + y = 9
-(2x + y = 5)
y is eliminated as it has reversed coefficients. Solve for x
3x + y – 2x – y = 9 – 5
x = 4
Substituting x in either of the equation to find y
2(4) + y = 5
8 + y – 8 = 5 – 8
y = -3
The solution is (4, -3)

Question 6.
$$\left\{\begin{array}{l}-x-2y=4 \\3x+2y=4\end{array}\right.$$
(________ , ________)

(4, -4)

Explanation:
$$\left\{\begin{array}{l}-x-2y=4 \\3x+2y=4\end{array}\right.$$
-x – 2y = 4
+(3x + 2y = 4)
y is eliminated as it has reversed coefficients. Solve for x
-x – 2y + 3x + 2y = 4 + 4
2x = 8
x = 8/2 = 4
Substituting x in either of the equation to find y
3(4) + 2y = 4
12 + 2y – 12 = 4 – 12
2y = -8
y = -8/2 = -4
The solution is (4, -4)

8.4 Solving Systems by Elimination with Multiplication

Solve each system of equations by multiplying first.

Question 7.
$$\left\{\begin{array}{l}x+3y=-2 \\3x+4y=-1\end{array}\right.$$
(________ , ________)

(1, -1)

Explanation:
$$\left\{\begin{array}{l}x+3y=-2 \\3x+4y=-1\end{array}\right.$$
Subtract the equations
3x + 9y = -6
-(3x + 4y = -1)
x is eliminated as it has reversed coefficients. Solve for y
3x + 9y – 3x – 4y = -6 + 1
5y = -5
y = -5/5
y = -1
Substituting y in either of the equation to find x
x + 3(-1) = -2
x – 3 = -2
x = -2 + 3
x = 1
The solution is (1, -1)

Question 8.
$$\left\{\begin{array}{l}2x+8y=22 \\3x-2y=5\end{array}\right.$$
(________ , ________)

(3, 2)

Explanation:
$$\left\{\begin{array}{l}2x+8y=22 \\3x-2y=5\end{array}\right.$$
Multiply equation 2 by 4 so that y can be eliminated
4(3x – 2y = 5)
12x – 8y = 20
2x + 8y = 22
+(12x – 8y = 20)
y is eliminated as it has reversed coefficients. Solve for x
2x + 8y + 12x – 8y = 22 + 20
14x = 42
x = 42/14
x = 3
Substituting y in either of the equation to find x
2(3) + 8y = 22
6 + 8y = 22
8y = 22 – 6
8y = 16
y = 16/8
y = 2
The solution is (3, 2)

8.5 Solving Special Systems

Solve each system. Tell how many solutions each system has.

ESSENTIAL QUESTION

Question 11.
What are the possible solutions to a system of linear equations, and what do they represent graphically?
Type below:
___________

A system of linear equations can have no solution, which is represented by parallel lines; one solution, which is represented by intersecting lines; and infinitely many solutions, which is represented by overlapping lines.

### Selected Response – Mixed Review – Page No. 266

Question 1.
The graph of which equation is shown?

Options:
A. y = −2x + 2
B. y = −x + 2
C. y = 2x + 2
D. y = 2x + 1

C. y = 2x + 2

Explanation:
Options A and B are eliminated as the slope of the graph is 2.
Option D is eliminated as the y-intercept from the graph should be 2.
Option C is the equation of the graph

Question 2.
Which best describes the solutions to the system
$$\left\{\begin{array}{l}x+y=-4 \\-2x-2y=0\end{array}\right.$$
Options:
A. one solution
B. no solution
C. infinitely many
D. (0, 0)

B. no solution

Explanation:
$$\left\{\begin{array}{l}x+y=-4 \\-2x-2y=0\end{array}\right.$$
Multiply equation 1 by 2 so that x can be eliminated
2(x + y = -4)
2x + 2y = -8
2x + 2y = -8
-2x – 2y = 0
x and y is eliminated
2x + 2y – 2x -2y = -8 + 0
0 = -8
The statement is false. Hence, the system has no solution.

Question 3.
Which of the following represents 0.000056023 written in scientific notation?
Options:
A. 5.6023 × 105
B. 5.6023 × 104
C. 5.6023 × 10-4
D. 5.6023 × 10-5

D. 5.6023 × 10-5

Explanation:
Move the decimal 5 points right to get the equation.
D. 5.6023 × 10-5

Question 4.
What is the solution to
$$\left\{\begin{array}{l}2x-y=1 \\4x+y=11\end{array}\right.$$
Options:
A. (2, 3)
B. (3, 2)
C. (-2, 3)
D. (3, -2)

A. (2, 3)

Explanation:
$$\left\{\begin{array}{l}2x-y=1 \\4x+y=11\end{array}\right.$$
2x – y = 1
4x + y = 11
y is eliminated as it has reversed coefficients. Solve for x.
2x – y + 4x + y = 1 + 11
6x = 12
x = 12/6 = 2
Substituting x in either of the equation to find y
4(2) + y = 11
8 + y = 11
y = 11 – 8
y = 3
The solution is (2, 3)

Question 5.
Which expression can you substitute in the indicated equation to solve
$$\left\{\begin{array}{l}3x-y=5 \\x+2y=4\end{array}\right.$$
Options:
A. 2y – 4 for x in 3x – y = 5
B. 4 – x for y in 3x – y = 5
C. 3x – 5 for y in 3x – y = 5
D. 3x – 5 for y in x + 2y = 4

D. 3x – 5 for y in x + 2y = 4

Explanation:
$$\left\{\begin{array}{l}3x-y=5 \\x+2y=4\end{array}\right.$$
Solve for y in equation 1
y = 3x – 5
Substitute in other equation x + 2y = 4

Question 6.
What is the solution to the system of linear equations shown on the graph?

Options:
A. -1
B. -2
C. (-1, -2)
D. (-2, -1)

C. (-1, -2)

Explanation:
The point of intersection is (-1, -2), which is the solution of the system

Question 7.
Which step could you use to start solving
$$\left\{\begin{array}{l}x-6y=8 \\2x-5y=3\end{array}\right.$$
Options:
A. Add 2x – 5y = 3 to x – 6y = 8.
B. Multiply x – 6y = 8 by 2 and add it to 2x – 5y = 3.
C. Multiply x – 6y = 8 by 2 and subtract it from 2x – 5y = 3.
D. Substitute x = 6y – 8 for x in 2x – 5y = 3.

C. Multiply x – 6y = 8 by 2 and subtract it from 2x – 5y = 3.

Explanation:
x – 6y = 8
2x – 5y = 3
Multiply the 1st equation by 2 so that the coefficient of variable x is the same in both equations
Subtract the equations as x has the same sign.

Question 8.
A hot-air balloon begins rising from the ground at 4 meters per second at the same time a parachutist’s chute opens at a height of 200 meters. The parachutist descends at 6 meters per second.
a. Define the variables and write a system that represents the situation.
Type below:
_____________

y represents the distance from the ground and x represents the time in seconds
y = 4x
y = -6x + 200

Question 8.
b. Find the solution. What does it mean?
Type below:
_____________