## Engage NY Eureka Math 8th Grade Module 4 Lesson 2 Answer Key

### Eureka Math Grade 8 Module 4 Lesson 2 Exercise Answer Key

Write each of the following statements in Exercises 1–12 as a mathematical expression. State whether or not the expression is linear or nonlinear. If it is nonlinear, then explain why.

Exercise 1.

The sum of a number and four times the number

Answer:

Let x be a number; then, x+4x is a linear expression.

Exercise 2.

The product of five and a number

Answer:

Let x be a number; then, 5x is a linear expression.

Exercise 3.

Multiply six and the reciprocal of the quotient of a number and seven.

Answer:

Let x be a number; then, 6∙\(\frac{7}{x}\) is a nonlinear expression. The expression is nonlinear because the number

\(\frac{7}{x}\)=7∙\(\frac{1}{x}\)=7∙x^{-1}. The exponent of the x is the reason it is not a linear expression.

Exercise 4.

Twice a number subtracted from four times a number, added to 15

Answer:

Let x be a number; then, 15+(4x-2x) is a linear expression.

Exercise 5.

The square of the sum of six and a number

Answer:

Let x be a number; then,(x+6)^{2} is a nonlinear expression. When you multiply(x+6)^{2}, you get x^{2}+12x+36. The x^{2} is the reason it is not a linear expression.

Exercise 6.

The cube of a positive number divided by the square of the same positive number

Answer:

Let x be a number; then, \(\frac{x^{3}}{x^{2}}\) is a nonlinear expression. However, if you simplify the expression to x, then it is linear.

Exercise 7.

The sum of four consecutive numbers

Answer:

Let x be the first number; then, x+(x+1)+(x+2)+(x+3) is a linear expression.

Exercise 8.

Four subtracted from the reciprocal of a number

Answer:

Let x be a number; then, \(\frac{1}{x}\)-4 is a nonlinear expression. The term \(\frac{1}{x}\) is the same as x^{-1}, which is why this expression is not linear. It is possible that a student may let x be the reciprocal of a number, \(\frac{1}{x}\), which would make the expression linear.

Exercise 9.

Half of the product of a number multiplied by itself three times

Answer:

Let x be a number; then, \(\frac{1}{2}\)∙x∙x∙x is a nonlinear expression. The term \(\frac{1}{2}\)∙x∙x∙x is the same as\(\frac{1}{2}\) x^{3}, which is why this expression is not linear.

Exercise 10.

The sum that shows how many pages Maria read if she read 45 pages of a book yesterday and \(\frac{2}{3}\) of the remaining pages today

Answer:

Let x be the number of remaining pages of the book; then, 45+\(\frac{2}{3}\) x is a linear expression.

Exercise 11.

An admission fee of $10 plus an additional $2 per game

Answer:

Let x be the number of games; then, 10+2x is a linear expression.

Exercise 12.

Five more than four times a number and then twice that sum

Answer:

Let x be the number; then, 2(4x+5) is a linear expression.

### Eureka Math Grade 8 Module 4 Lesson 2 Problem Set Answer Key

Students practice writing expressions and identifying them as linear or nonlinear.

Write each of the following statements as a mathematical expression. State whether the expression is linear or nonlinear. If it is nonlinear, then explain why.

Question 1.

A number decreased by three squared

Answer:

Let x be a number; then, x-3^{2} is a linear expression.

Question 2.

The quotient of two and a number, subtracted from seventeen

Answer:

Let x be a number; then, 17-\(\frac{2}{x}\) is a nonlinear expression. The term \(\frac{2}{x}\) is the same as 2∙\(\frac{1}{x}\) and \(\frac{1}{x}\)=x^{-1}, which is why it is not linear.

Question 3.

The sum of thirteen and twice a number

Answer:

Let x be a number; then, 13+2x is a linear expression.

Question 4.

5.2 more than the product of seven and a number

Answer:

Let x be a number; then, 5.2+7x is a linear expression.

Question 5.

The sum that represents the number of tickets sold if 35 tickets were sold Monday, half of the remaining tickets were sold on Tuesday, and 14 tickets were sold on Wednesday

Answer:

Let x be the remaining number of tickets; then, 35+\(\frac{1}{2}\) x+14 is a linear expression.

Question 6.

The product of 19 and a number, subtracted from the reciprocal of the number cubed

Answer:

Let x be a number; then, \(\frac{1}{x^{3}}\) -19x is a nonlinear expression. The term \(\frac{1}{x^{3}}\) is the same as x^{-3}, which is why it is not linear.

Question 7.

The product of 15 and a number, and then the product multiplied by itself four times

Answer:

Let x be a number; then,(15x)^{4} is a nonlinear expression. The expression can be written as 15^{4}∙x^{4}. The exponent of 4 with a base of x is the reason it is not linear.

Question 8.

A number increased by five and then divided by two

Answer:

Let x be a number; then, \(\frac{x+5}{2}\) is a linear expression.

Question 9.

Eight times the result of subtracting three from a number

Answer:

Let x be a number; then, 8(x-3) is a linear expression.

Question 10.

The sum of twice a number and four times a number subtracted from the number squared

Answer:

Let x be a number; then, x^{2}-(2x+4x) is a nonlinear expression. The term x^{2} is the reason it is not linear.

Question 11.

One-third of the result of three times a number that is increased by 12

Answer:

Let x be a number; then, \(\frac{1}{3}\) (3x+12) is a linear expression.

Question 12.

Five times the sum of one-half and a number

Answer:

Let x be a number; then, 5(\(\frac{1}{2}\)+x) is a linear expression.

Question 13.

Three-fourths of a number multiplied by seven

Answer:

Let x be a number; then, \(\frac{3}{4}\)x∙7 is a linear expression.

Question 14.

The sum of a number and negative three, multiplied by the number

Answer:

Let x be a number; then, (x+(-3))x is a nonlinear expression because (x+(-3))x=x^{2}-3x after using the distributive property. It is nonlinear because the power of x in the term x^{2} is greater than 1.

Question 15.

The square of the difference between a number and 10

Answer:

Let x be a number; then, (x-10)^{2} is a nonlinear expression because (x-10)^{2}=x^{2}-20x+100. The term x^{2} is a positive power of x>1; therefore, this is not a linear expression.

### Eureka Math Grade 8 Module 4 Lesson 2 Exit Ticket Answer Key

Write each of the following statements as a mathematical expression. State whether the expression is a linear or nonlinear expression in x.

Question 1.

Seven subtracted from five times a number, and then the difference added to nine times a number

Answer:

Let x be a number; then, (5x-7)+9x. The expression is a linear expression.

Question 2.

Three times a number subtracted from the product of fifteen and the reciprocal of a number

Answer:

Let x be a number; then, 15∙\(\frac{1}{x}\)-3x. The expression is a nonlinear expression.

Question 3.

Half of the sum of two and a number multiplied by itself three times

Answer:

Let x be a number; then, \(\frac{1}{2}\) (2+x^{3}). The expression is a nonlinear expression.