System of 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. 

How to Solve Linear Equations with Two Variables?

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.

1. 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. 

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Solve the system of linear equations step by step

This calculator will solve the system of linear equations of any kind, with steps shown, using either the Gauss-Jordan elimination method, the inverse matrix method, or Cramer's rule.

Related calculator: System of Equations Calculator

Comma-separated, for example, x+2y=5,3x+5y=14.

Leave empty for autodetection or specify variables like x,y (comma-separated).

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Your Input

Solve $$$\begin{cases} 5 x - 2 y = 1 \\ x + 3 y = 7 \end{cases}$$$ for $$$x$$$, $$$y$$$ using the Gauss-Jordan Elimination method.

Solution

Write down the augmented matrix: $$$\left[\begin{array}{cc|c}5 & -2 & 1\\1 & 3 & 7\end{array}\right]$$$.

Perform the Gauss-Jordan elimination (for steps, see Gauss-Jordan elimination calculator): $$$\left[\begin{array}{cc|c}5 & -2 & 1\\0 & \frac{17}{5} & \frac{34}{5}\end{array}\right]$$$.

Back-substitute:

$$$y = \frac{\frac{34}{5}}{\frac{17}{5}} = 2$$$

$$$x = \frac{1 - \left(-2\right) \left(2\right)}{5} = 1$$$

Answer

$$$x = 1$$$A

$$$y = 2$$$A

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