A Simple Approach to Solving Systems of Linear Equations


The whole process of solving a system of two linear equations with two variables can be broken down in the following process:

Step 1: Get both equations in slope-intercept form. This will allow us to compare the two equations quickly and easily.

Step 2: If they have the same slope and different y-intercepts, they are parallel and there are no solutions. If they have the same slope and the same y-intercepts, they are the same line and have infinite solutions. If they do not have the same slope, move on to step 3.

Step 3: Set the two equations equal to each other and solve for x. This step requires an example. Suppose we have the two equations y = 3x + 4 and y = -2x + 1. Then we also know that 3x + 4 = -2x + 1 since they both equal y, and we can quickly solve for x.

Step 4: Plug in the value you found for x into one of the two equations and solve for y.

Step 5: Check your answer by plugging your values for x and y into both equations to see if you get something that makes sense.

With that out of the way, let’s look at an example. Suppose we have 2x + 3y = 12 and -4x + 2y = -8. First, we put each equation into slope-intercept form as follows:

2x + 3y = 12 
3y = -2x + 12 
y = -\frac{2}{3}x + 4 

and

-4x + 2y = -8 
2y = 4x - 8 
y = 2x - 4 

We see that the slopes are different, so these equations can’t be parallel or the same line. Now we go on to set these two equations equal to each other and solve for x.

2x - 4 = -\frac{2}{3}x + 4 
2x + \frac{2}{3}x = 8 
\frac{6}{3}x + \frac{2}{3}x = 8 
\frac{8}{3}x = 8 
8x = 24 
x = 3 

Now let’s put in 3 for x and see what we get for y:

2x + 3y = 12
2(3) + 3y = 12
6 + 3y = 12
3y = 12
y = 2

So now we think the solution for the system is x = 3 and y = 2, or just (3, 2). To verify this, we should plug in 3 for x and 2 for y into both equations to see what we get. We’ll leave this verification to the reader as an exercise.