This calculator solves Systems of Linear Equations using Gaussian Elimination Method, Inverse Matrix Method, or Cramer's rule. Also you can compute a number of solutions in a system of linear equations (analyse the compatibility) using Rouché–Capelli theorem.
Enter coefficients of your system into the input fields. Leave cells empty for variables, which do not participate in your equations. To input fractions use /: 1/3.
- 2x-2y+z=-3 x+3y-2z=1 3x-y-z=2
- Leave extra cells empty to enter non-square matrices.
- You can use decimal (finite and periodic) fractions: 1/3, 3.14, -1.3(56), or 1.2e-4; or arithmetic expressions: 2/3+3*(10-4), (1+x)/y^2, 2^0.5 (=2), 2^(1/3), 2^n, sin(phi), or cos(3.142rad).
- Use ↵ Enter, Space, ←↑↓→, ⌫, and Delete to navigate between cells, Ctrl⌘ Cmd+C/Ctrl⌘ Cmd+V to copy/paste matrices.
- Drag-and-drop matrices from the results, or even from/to a text editor.
- To learn more about matrices use Wikipedia.
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- \begin{pmatrix}9&2&-4\\b+a&9&7\\0&c&8\end{pmatrix}=\begin{pmatrix}9&a&-4\\7&9&7\\0&16&8\end{pmatrix}
- \begin{pmatrix}4&0\\6&-2\\3&1\end{pmatrix}=\begin{pmatrix}x&0\\6&y+4\\\frac{z}{3}&1\end{pmatrix}
- x+\begin{pmatrix}3&2\\1&0\end{pmatrix}=\begin{pmatrix}6&3\\7&-1\end{pmatrix}
- 2\begin{pmatrix}1&2\\0&1\end{pmatrix}x+\begin{pmatrix}3&4\\2&1\end{pmatrix}=\begin{pmatrix}1&2\\3&4\end{pmatrix}
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Sorry, your browser does not support this applicationExamples
- x+y+z=25,\:5x+3y+2z=0,\:y-z=6
- x+2y=2x-5,\:x-y=3
- 5x+3y=7,\:3x-5y=-23
- x^2+y=5,\:x^2+y^2=7
- xy+x-4y=11,\:xy-x-4y=4
- 3-x^2=y,\:x+1=y
- xy=10,\:2x+y=1
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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