Solving Systems of (Linear) Equations

To solve a system of simultaneous linear equations, there must be as many equations as there are variables:

$\begin{aligned} 12 x + 4 y & = 36 \\ 6 x - 3 y & = 3 \end{aligned}$

There are two methods we can use to solve this system:

Substitution

First, we take one equation and solve for one of the variables. Here, we take the first, and solve for $x$ :

$\begin{aligned} 12 x + 4 y & = 36 \\ 12 x & = 36 - 4 y \\ x & = 3 - \frac{1}{3} y \end{aligned}$

Now we take this value and plug it into the other equation:

$\begin{aligned} 6 x - 3 y & = 3 \\ 6 (3 - \frac{1}{3} y) - 3 y & = 3 \\ 18 - 2 y - 3 y & = 3 \\ 18 - 5 y & = 3 \\ - 5 y & = - 15 \\ y & = 3 \end{aligned}$

Now that we know the value of one variable, plug it into either of the original equations to solve for the value of the other variable:

$\begin{aligned} 12 x + 4 y & = 36 \\ 12 x + 4 (3) & = 36 \\ 12 x + 12 & = 36 \\ 12 x & = 24 \\ x & = 2 \end{aligned}$

We should verify that our variables are correct, so plug $x$ and $y$ into each equation and make sure it is true. Let’s start with the first equation:

$\begin{aligned} 12 x + 4 y & = 36 \\ 12 (2) + 4 (3) & = 36 \\ 24 + 12 & = 36 ✓ \end{aligned}$

We can do the same with the other equation:

$\begin{aligned} 6 x - 3 y & = 3 \\ 6 (2) - 3 (3) & = 3 \\ 12 - 9 & = 3 ✓ \end{aligned}$

Elimination

We will multiply the equations by constants to make the coefficients of one variable equal. Here, let us try to make the coefficients in front of each equation’s $x$ equal.

$\begin{aligned} [12 x + 4 y & = 36] * 6 \\ [6 x - 3 y & = 3] * 12 \end{aligned}$

To do so, we will multiply the first equation by 6, and the second equation by 12. (We multiply each equation by the coefficient in front of the other equation’s $x$ variable):

$\begin{aligned} 72 x + 24 y & = 216 \\ 72 x - 36 y & = 36 \end{aligned}$

Now we subtract the second equation from the first, which should get rid of $x$ :

$\begin{aligned} 72 x + 24 y & = 216 \\ - [72 x - 36 y & = 36] \end{aligned}$

Be careful to distribute the minus sign carefully:

$\begin{aligned} [72 x - 72 x] + [24 y - (- 36 y)] & = [216 - 36] \\ 60 y & = 180 \\ y & = 3 \end{aligned}$

Now that we have the value of one variable, plug it in to either equation.

$\begin{aligned} 12 x + 4 y & = 36 \\ 12 x + 4 (3) & = 36 \\ 12 x + 12 & = 36 \\ 12 x & = 24 \\ x & = 2 \end{aligned}$

Now that we have both variables, we can plug them in to each equation to double check them, same as before.