How to Solve Systems of Equations in Seven Steps

Introduction

Solving systems of linear equations is really hard for many students because it's simply not taught in a way that students can understand. The average curriculum teaches a few "methods" for how solve these systems, but these methods can't really be applied to more complicated systems. Moreover, these "methods" are really cookie-cutter in nature, and require perfectly-formulated examples to work well. Instead of following the "traditional" method of teaching this subject, we have a much easier way to solve this stuff.

Step 1: Get Both Equations in Slope-Intercept Form

This method requires that both of your equations be written in slope-intercept form, which looks something like y = mx + b. Many of your problems and exercises will give you equations in other formats, like the standard form that looks like Ax + Bx = C. Before you can solve a system of equations, however, you must check to make sure that the lines in question aren't parallel (which means there are no solutions) and that they aren't actually the same line (which means there are infinite solutions). To do this, we need the equations in the slope-intercept form.

Step 2: Compare the Slopes and Y-Intercepts

If the slopes of the two equations are different, then there is exactly one solution to the system of equations. If the slopes are the same, then there will not be exactly one solution.

When you have slopes that are the same, you have to look at the y-intercepts of each equation to see if there are zero or infinite solutions. If the y-intercepts are the same, then the two equations are the same (which should be obvious), and there are infinite solutions. Otherwise, there will be no solutions because the lines are parallel.

This much should have been a review for most students. Now that we've verified that there is one solution, it's time to find that solution with a simple method!

How Many Solutions?

A brief description of how to compare two equations to see how many solutions there are.
A brief description of how to compare two equations to see how many solutions there are.

Step 3: Set the Equations Equal to Each Other

If your equations were y = 3x + 4 and y = -2x + 1, then you can have 3x + 4 = -2x + 1 because both of these expressions are equal to y. This is a very basic type of substitution that's really easy to perform once you have the equations in slope-intercept form.

The point here is that normally you're taught to use "elimination", "substitution", or "graphing" to solve systems of equations. The problem is that "elimination" is very difficult to use unless the problems are hand-picked, and "graphing" is very difficult to do unless nice, even numbers are used. In the real world, you need a substitution method that's easy to do and is resistant to mistakes. Since you have to put the equations in slope-intercept form anyway, this method will always be available, and will always be very simple to follow.

Step 4: Solve for X

We had y = 3x + 4 and y = -2x + 1 in our previous example, and that gave us 3x + 4 = -2x + 1. What we're left with here is just a simple linear equation of a type that that we've solved over and over again. For example, we could have the following:

3x + 4 = -2x + 1
5x + 4 = 1
5x = -3
x = -3/5

And we have solved for x in just a few simple steps. Doesn't this beat the more "traditional" methods?!

Step 5: Solve for Y

We said that we found x = -3/5 in step 4, so now we'll just need to substitute that into one of our starting equations to find a value for y. We have y = 3x + 4, so here we go:

y = 3(-3/5) + 4
y = -9/5 + 4
y = -9/5 + 20/5
y = 11/5

And that took all of maybe a minute to do if you're familiar with how to handle the fractions, which you should be at this point in learning Algebra 1.

Step 6: Check Your Answer

Take the x and y coordinate that you get as a solution for this problem and plug it into each of the equations, making sure that it works. It's not enough to know how to get *an* answer, but you want to be able to make sure that you have *the* answer by knowing ways to quickly and easily check yourself. Taking a moment to check your answers using easy methods like substituting your answer into the original equations will give your grade a dramatic boost.

Step 7: Laugh at Your Classmates

Now that you have the answer, and you're sure that it's the right answer, it's time to sit back and watch everyone else work really hard and get confused over a problem that you just finished with ease!

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