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:


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:

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


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.