What is the solution to the system of equations calculator

Step-by-Step Examples

Algebra

Solve by Substitution Calculator

Step 1:

Enter the system of equations you want to solve for by substitution.

The solve by substitution calculator allows to find the solution to a system of two or three equations in both a point form and an equation form of the answer.

Step 2:

Click the blue arrow to submit.

A system of linear equations consists of multiple linear equations. Linear equations with two variables correspond to lines in the coordinate plane, so this linear equation system is nothing more than asking if, and if yes, where the two lines intersect. This implies it can have no solution (if the lines are parallel), one solution (if they intersect) or infinitely many solutions (if the lines are equal).

There are three important ways to solve such systems: by insertion, by equalization and by adding.

Equalization means you solve both equations for the same variable and then equalize them. This means, one variable remains and the calculation is then easy.

Equation systems

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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).

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

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

This solver (calculator) will try to solve a system of 2, 3, 4, 5 equations of any kind, including polynomial, rational, irrational, exponential, logarithmic, trigonometric, hyperbolic, absolute value, etc. It can find both the real and the complex solutions. To solve a system of linear equations with steps, use the system of linear equations calculator.

Enter a system of equations:

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

Solve for (comma-separated):

Leave empty for automatic determination, or specify variables like x,y.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

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