# 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:

$$3x + 2y = 7$$ $$2x + 3y = 8$$

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.

$$2 * (3x + 2y = 7) => 6x + 4y = 14$$ $$3 * (2x + 3y = 8) => 6x + 9y = 24$$

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:

$$x = \frac{7-2y}{3}$$

Next we can substitute that value for $$x$$ into the second equation:

$$2(\frac{7-2y}{3}) + 3y = 8$$

We can now easily solve for $$y$$.