# Simultaneous Equations

"Simultaneous equations, also known as a system of equations, is a finite set of equations for which common solutions are sought." - Wikipedia.

Linear algebra can be used to solve a system of linear equations.

## Solving By Hand

### Elimination

Let's say we have the following equations:

\begin{equation} 3x + 2y = 7 \end{equation} \begin{equation} 2x + 3y = 8 \end{equation}We can find the values of \(x\) and \(y\) by removing a variable through elimination.

Let's first make the leading coefficient the same for one of the variables.

\begin{equation} 2 * (3x + 2y = 7) => 6x + 4y = 14 \end{equation} \begin{equation} 3 * (2x + 3y = 8) => 6x + 9y = 24 \end{equation}Here we're trying to make the coefficient for \(x\) the same in both equations. Notice that we multiply by the coefficient of the other equation.

We then subtract the equations to eliminate \(x\):

6x + 4y = 14 -(6x + 9y = 24) ________________ -5y = -10

We can now easily solve for **.

### Substitution

Using the same equations as above, we can also solve the equations by substituting a value from one into the other.

First, let's get \(x\) on one side in the first equation:

\begin{equation} x = \frac{7-2y}{3} \end{equation}Next we can substitute that value for \(x\) into the second equation:

\begin{equation} 2(\frac{7-2y}{3}) + 3y = 8 \end{equation}We can now easily solve for \(y\).