# System of Equations Calculator

System of equations calculator helps to determine the value of the variables in a given set of equations. When one or more linear equations are used to find the values of a set of variables it is known as a system of linear equations. These are also termed simultaneous equations.

## What is System of Equations Calculator?

System of Equations Calculator is an online tool that helps to find the value of the three variables, x,y,z, through the given equations. Many computational algorithms are based on the system of linear equations. These have a widespread applications in Engineering, Physics, Chemistry, etc. To use the **system of equations calculator**, enter the values in the input boxes.

### System of Equations Calculator

**NOTE:** Enter numbers upto 3 digits only.

## How to Use the System of Equations Calculator?

Please follow the below steps to find the values of the variables using the system of equations calculator:

**Step 1:**Go to Cuemath's online system of equations calculator.**Step 2:**Enter the values in the input boxes.**Step 3:**Click on the**"Solve"**button to find the x, y, z.**Step 4:**Click on the**"Reset"**button to clear the fields and enter new values.

## How Does System of Equations Calculator Work?

There are many methods to solve a system of linear equations. One of the commonly employed methods is to take the help of matrices for solving simultaneous equations. This is popularly known as the Cramer's rule. Given below are the steps to solve a system of equations using this technique.

Suppose the system of equations is given by:

\(a_{1}x + b_{1}y + c_{1}z = d_{1}\)

\(a_{2}x + b_{2}y + c_{2}z = d_{2}\)

\(a_{3}x + b_{3}y + c_{3}z = d_{3}\)

**Step 1:** Find the different determinants as below:

D = \(\begin{vmatrix} a_{1} &b_{1} & c_{1}\\ a_{2} &b_{2} & c_{2}\\ a_{3} &b_{3} & c_{3} \end{vmatrix}\)

\(D_{x}\) = \(\begin{vmatrix} d_{1} &b_{1} & c_{1}\\ d_{2} &b_{2} & c_{2}\\ d_{3} &b_{3} & c_{3} \end{vmatrix}\)

\(D_{y}\) = \(\begin{vmatrix} a_{1} &d_{1} & c_{1}\\ a_{2} &d_{2} & c_{2}\\ a_{3} &d_{3} & c_{3} \end{vmatrix}\)

\(D_{z}\) = \(\begin{vmatrix} a_{1} &b_{1} & d_{1}\\ a_{2} &b_{2} & d_{2}\\ a_{3} &b_{3} & d_{3} \end{vmatrix}\)

**Step 2 :** Using these determinants find the value of the variables as follows:

x = \(D_{x}\) / D

y = \(D_{y}\) / D

z = \(D_{z}\) / D

To simplify the process of finding the value of each determinant, row and column transformations can be used.

## Solved Examples on System of Equations Calculator

**Example 1:**

Find the values of x, y, z in the given equation: \( \begin{matrix} 5x\;+\;2y\;+\;1z \;=\; 4\\ 2x\;+\;3y\;+\;1z \;=\; 4\\ 2x\;+\;2y\;+\;1z \;=\; 2\end{matrix} \)

**Solution:**

D = \(\begin{vmatrix} 5 & 2 & 1\\ 2 & 3 & 1\\ 2 & 3 & 1 \end{vmatrix}\)

\(D_{x}\) = \(\begin{vmatrix} 4 & 2 & 1\\ 4 & 3 & 1\\ 2 & 3 & 1 \end{vmatrix}\)

\(D_{y}\) = \(\begin{vmatrix} 5 & 4 & 1\\ 2 & 4 & 1\\ 2 & 2 & 1 \end{vmatrix}\)

\(D_{z}\) = \(\begin{vmatrix} 5 & 2 & 4\\ 2 & 3 & 4\\ 2 & 2 & 2 \end{vmatrix}\)

x = \(D_{x}\) / D = 0.667

y = \(D_{y}\) / D = 2

z = \(D_{z}\) / D = -3.333

**Example 2:**

Find the values of x, y, z in the given equation: \( \begin{matrix} 1x\;+\;1y\;-\;1z \;=\; 6\\ 3x\;-\;2y\;+\;1z \;=\; -5\\ 1x\;+\;3y\;-\;2z \;=\; 14\end{matrix} \)

**Solution:**

D = \(\begin{vmatrix} 1 & 1 & -1\\ 3 & -2 & 1\\ 1 & 3 & -2 \end{vmatrix}\)

\(D_{x}\) = \(\begin{vmatrix} 6 & 1 & -1\\ -5 & -2 & 1\\ 14 & 3 & -2 \end{vmatrix}\)

\(D_{y}\) = \(\begin{vmatrix} 1 & 6 & -1\\ 3 & -5 & 1\\ 1 & 15 & -2 \end{vmatrix}\)

\(D_{z}\) = \(\begin{vmatrix} 1 & 1 & 6\\ 3 & -2 & -5\\ 1 & 3 & 14 \end{vmatrix}\)

x = \(D_{x}\) / D = 1

y = \(D_{y}\) / D = 3

z = \(D_{z}\) / D = 2

Similarly, you can try the system of equations calculator and find the values of x,y, z

- \( \begin{matrix} 2x\;+\;2y\;+\;5z \;=\; 4\\ x\;+\;3y\;+\;1z \;=\; 4\\ 2x\;+\;4y\;+\;1z \;=\; 2\end{matrix} \)
- \( \begin{matrix} x\;-\;3y\;+\;7z \;=\; 13\\ x\;+\;y\;+\;z \;=\; 1\\ x\;-\;2y\;+\;3z \;=\; 4\end{matrix} \)

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