Previously we have learned the Introduction to Simultaneous Linear Equations. Kids should know that in linear equations in one variable, only one variable is introduced whose value has to solve using simple mathematical operations like +,-,/, and *.
But the theory of the linear equations remains unchanged in the case of linear equations in two variables also. Explore this guide and find detailed information on the Solution of a Linear Equation in Two Variables definition, how it works, what are the methods used, word problems, solved questions and answers, etc.
Also Check:
- Solvability of Linear Simultaneous Equations
- Word Problems on Linear Equations
- Problems on Linear Equations in One Variable
Solution of a Linear Equation in Two Variables Definition
The solution of linear equations in two variables, ax+by = c, is a specific point in the graph so that when x-coordinate is multiplied by a and y-coordinate is multiplied by b, then the sum of these two values will result as c. Basically, there are infinitely many solutions for a linear equation in two variables.
Linear Equation in Two Variable Formula
If a, b, and c are real numbers and if they both are equal to 0, then ax + by = c is known as a linear equation in two variables.
Where x and y are represented as two variables.
The numbers a and b are called the coefficients of the equation ax+by = c.
The number c is called the constant of the equation ax + by = c.
Example: 3x + 2y = 5 and -4x + 5y = 10 are linear equations in two variables.
How to Find the Solution of a Linear Equation in Two Variables?
Generally, to solve the equations involving two variables there is a possibility to use two methods. They are:
1. Substitution Method:
In this method, we have to follow some basic steps to solve the linear equations in two variables. Actually, we know that linear equations with two variables need at least two equations. Here, we discover the value of any one variable from the given expressions and substitute the result in the second equation to calculate the value of the variable. Now, apply the value of a variable in another equation and find the value of the other variables.
Look at the solved 2-variable equations examples mentioned below for a better understanding of the concept using the substitution method.
2. Elimination Method:
The method of determining the variables from the equations including two unknown quantities by eliminating one of the variables and then answering the resulting equation to find the value of one variable and then applying it into any one of the equations to obtain the other variable is called a method of elimination. By multiplying both equations with a number that any of the coefficients may have a multiple common, in such a way the elimination process is completed.
For getting a good grip on this method, let’s take a look at the problems of solving linear equations using the elimination method.
Graphical Method for Solving Linear Equations in Two Variables
Following are the necessary steps that included in solving linear equations in two variables graphically:
- First, we have to graph each equation to solve the system of equations graphically.
- Next, learn how to graph an equation manually. Just covert it to the form y = mx+b by solving the equation for y.
- Now, put the values of x as 0, 1, 2, and so on and calculate the corresponding values of y, or vice-versa.
- Determine the point where both lines intersect.
- Finally, the point of intersection is the solution of the provided system.
Example: Consider a system of two linear equations; a1 + b1 + c1 = 0 and a2 + b2 + c2 = 0.
Problems on Solution of Linear Equations in Two Variables via Substitution & Elimination Methods | 2 Variable Equations Examples
Example 1:
Find the value of variables that satisfies the following equation:
x + 2y = 10 ……………… (i)
2x + y = 20 ……………… (ii)
Solution:
By using the substitution method, solve the pair of linear equations, we have:
x + 2y = 10 ……………… (i)
2x + y = 20 ……………… (ii)
From equation (ii), we get
y = 21 – 2x
Substitute the value of y in equation (i):
x + 2y = 10
x + 2(20-2x) = 10
x + 40 – 4x = 10
3x + 40 = 10
3x = 10 – 40
3x = -30
x = -30/3
x = -3
Substituting the x =-3 in equation (ii):
2x + y = 20
2(-3) + y = 20
-6 + y = 20
y = 20+6
y = 26
Hence x = -3 and y = 26 are the solutions of given linear equations in two variables via the substitution method.
Example 2:
Find the values of x and y by solving the given linear equations in two variables using elimination method:
3x + 4y = 10 ………. (i)
x + y = 20 ……….. (ii)
Solution:
Given equations are:
3x + 4y = 10 ………. (i)
x + y = 20 ……….. (ii)
Now, start with Multiplying equation (ii) by 3, we get;
3x + 3y = 60 ……….(iii)
Subtract (i) from (iii), we get
4y – 3y = 10 – 60
y = -50
Subsitute the y value in (ii), we get
x – 50 = 20
x = 20 + 50
x = 70
Therefore, x = 70 and y = -50 are the solutions of a linear equation in two variables.
FAQs on Linear Equations in Two Variables Questions
1. What are the different methods used to solve Linear Equations in Two Variables?
There are five methods used for finding the solutions of a linear equation in two variables. They are as follows:
- Substitution Method
- Cross Multiplication Method
- Elimination Method
- Graphical Method
- Determinant Method
2. How to find a solution of a linear equation in two variables?
3. What is the solution of the linear equation?
A point at which the planes or lines representing the linear equations cross each other is called the solution of a linear equation.
4. How many solutions are there for linear equations in two variables?
The answer for linear equations in two variables has how many solutions question is stated here. They are infinitely many solutions.