Solving Systems of (Linear) Equations

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

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

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 \(x\):

\[\begin{align*} 12x+4y&=36\\ 12x&=36-4y\\ x&=3-\frac{1}{3}y\\ \end{align*}\]

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

\[\begin{align*} 6x-3y&=3\\ 6(3-\frac{1}{3}y)-3y&=3\\ 18-2y-3y&=3\\ 18-5y&=3\\ -5y&=-15\\ y&=3\\ \end{align*}\]

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:

\[\begin{align*} 12x+4y&=36\\ 12x+4(3)&=36\\ 12x+12&=36\\ 12x&=24\\ x&=2\\ \end{align*}\]

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:

\[\begin{align*} 12x+4y&=36\\ 12(2)+4(3)&=36\\ 24+12&=36 \checkmark \\ \end{align*}\]

We can do the same with the other equation:

\[\begin{align*} 6x-3y&=3\\ 6(2)-3(3)&=3\\ 12-9&=3 \checkmark \\ \end{align*}\]

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.

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

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):

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

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

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

Be careful to distribute the minus sign carefully:

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

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

\[\begin{align*} 12x+4y&=36\\ 12x+4(3)&=36\\ 12x+12&=36\\ 12x&=24\\ x&=2\\ \end{align*}\]

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