Mathematics

System of equations - formula in LaTeX

A system of two equations joined by a brace on the left. The cases environment aligns rows to the left. If the equals signs must line up exactly, use aligned inside \left\{ and \right. instead, with & before each equals sign. Both need the amsmath package.

\begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}

LaTeX code

Open in editor

$$
\begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}
$$

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What the symbols mean

$x, y$: unknowns
$a_1, b_1, c_1$: coefficients of the first equation
$a_2, b_2, c_2$: coefficients of the second equation

A math formula - one of the most frequently typed in assignments, lecture notes and calculus or algebra homework.

Worked example

Adding the equations cancels y, which gives x = 2 straight away and y = 1 from the second equation. The solution sits on the same line as the system, after an arrow.

\begin{cases} 2x + y = 5 \\ x - y = 1 \end{cases} \quad\Rightarrow\quad 3x = 6, \quad x = 2, \quad y = 1

Other ways to write it

Equals signs lined up exactly; \right. is an invisible right bracket.
$\left\{\begin{aligned} 2x + y &= 5 \\ x - y &= 1 \end{aligned}\right.$
```
\left\{\begin{aligned} 2x + y &= 5 \\ x - y &= 1 \end{aligned}\right.
```
Three equations in three unknowns: just add another row.
$\begin{cases} x + y + z = 6 \\ x - y = -1 \\ z - x = 2 \end{cases}$
```
\begin{cases} x + y + z = 6 \\ x - y = -1 \\ z - x = 2 \end{cases}
```
The same system in matrix form Ax = b.
$\begin{pmatrix} a_1 & b_1 \\ a_2 & b_2 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} c_1 \\ c_2 \end{pmatrix}$
```
\begin{pmatrix} a_1 & b_1 \\ a_2 & b_2 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} c_1 \\ c_2 \end{pmatrix}
```

LaTeX commands in this formula

\begin{cases}: cases
x_{0}: subscript; wrap several characters in braces

How to use this formula in LaTeX?

Copy the code with the button above - $$…$$ is centered display math.
Paste it into the CamteX editor, Overleaf or any LaTeX document.
Inside a sentence, use single $…$ instead of double.
The formula needs the amsmath package - CamteX templates include it in the preamble.

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