Linear Equations in Two Variables Calculator

The definition of a linear equation in two variables is an equation written in the form of ax + by + c = 0, where a, b, c are real numbers and a, b are also coefficients of x and y which are not equal to 0. The result of such equations is an x & y value which makes two sides of an equation equal. 

There are various ways to find out the linear equation in two variables. Here we are going to explain two methods to solve variables of linear equations. They are the following ones: 

• Method of Substitution
• Method of Elimination

1. Substitution Method:

One of the commonly used methods to solve linear equations is the substitution method. By using this approach, you will get the result of one variable by substituting the given inputs in one equation. After that, you have to substitute the result in the other equations and solve the other variable value. For better understanding, kindly look at the below solved 2-variable equations example which is calculated using the substitution method.

Example:

Solve x + y = 4 and x + 2y = 6

Solution:

Given linear equations are 

x + y = 4 ……….(1)

x + 2y = 6 ……...(2)

From (1), x = 4 - y ……..(3)

Substitute (3) in (2), 

x + 2y = 6

4 - y + 2y = 6

4 + y = 6

Subtract 4 on both sides of the equation

4 + y - 4 = 6 - 4

Y = 2 ………(4)

Substitute (4) in (1)

x + y = 4

x + 2 = 4

Subtract 2 on both sides of the equation

x + 2 - 2 = 4 - 2

X = 2

Hence, x = 2 and y = 2 are the variable values for the given linear equations.

2. Elimination Method:

The procedure to solve the linear equation in two variables using the elimination method is explained here in a detailed manner. The objective is to make the coefficients of one variable equal to the same variable of the other equation. The elimination of the same variables can be done by either adding or subtracting one from another.

Practice solving linear equations with two variables via the method of elimination through example and online calculators and get a good grip on it.

Example: 

Solve the system of equations: 2x + 7y = 10 and 3x + y = 6.

Solution:

Let’s consider the equations:

2x + 7y = 10…………….. (1)

3x + y = 6………………… (2)

To make the coefficients of one variable similar to each other, we are multiplying equation (2) with 7 then, 

2x + 7y = 10
(3*7)x + 7y = 6*7

21x + 7y = 32

Now, subtract equation (1) with equation (2), we get

19x = 32

x= 32/19

Substitute the value of x in equation (1), 

2(32/19) + 7y = 10

64/19 + 7y = 10

7y = 10 - 64/19

7y = 126/19

y = 18/19

Hence, x = 32/19 and y = 18/19. 

1. How can I solve the System of Linear Equations in Two Variables?

By using various methods we can easily solve the System of Linear Equations in Two Variables. They are as follows:

• Substitution Method
• Graphical Method
• Elimination Method
• Cross Multiplication Method
• Determinant or Matrix Method

2. How many solutions do have Linear Equations with Two Variables?

Let’s assume that you have a1x + b1y + c1= 0 and a2x + b2y + c2 = 0 the the solutions of a linear equation with two variables are: 

• One and unique if a1a2 ≠ b1b2
• None if a1a2 ≠ b1b2 ≠ c1c2
• Infinitely many if a1a2 = b1b2 = c1c2

3. Can I Solve Linear Equations in Two Variables in a fraction of seconds?

Yes, you can easily & quickly solve linear equations in two variables by using Linearequationscalculator.com offered free online & handy Linear Equations in Two Variables Calculator.