What's a System?
A system of equations is a set of 2 or more equations. The systems you study in Algebra 1 generally consist of two linear equations. The graphs of linear equations are straight lines, so the goal is to figure out the point where the two lines cross. This ordered pair will be the solution to the system. The solution to a system is the point that works for all of the equations in the system.
When your system has two linear equations, there are 3 things that can happen.
1) The lines will cross at one point. This means there will be one solution to the system.
2) The lines will be parallel and never cross. This means there is no solution.
3) The lines will be the same and lay right on top of each other. If this happens, there will be infinitely many solutions because each point on the line works for both equations.
In this lesson, we'll focus on systems that have one solution. The system shown below consists of two linear equations: y = 2x + 3 and y = -3x + 13.
You can see from the graph that the two lines cross at the point (2,7). This means that the solution to this system is (2,7). If you plug in 2 for x and 7 for y, they work in both equations.
You can always graph the equations in a system to find the solution point. However, it can be difficult to tell where the lines cross if the answer is very large or if the answer involves fractions. The substitution method is an alternative to graphing. It's a method that can be used to solve a system without graph paper and you'll get an exact answer every time.
Open the next tab to see how to use the substitution method.
What is the Substitution Method?
Ever had a substitute teacher? When the regular teacher is gone, you get a substitute, or replacement, teacher in his or her place. It's the same basic idea here with systems. One thing is taken out and a substitute, or replacement, is put in its place.
Solve. y = -2x + 5 and x = 1
This is a system with two linear equations. Remember, our goal is to figure out the point where the two lines cross. In this system, we already have half of the answer since we know that x = 1. All that's left to do is to find y.
We know that x = 1, so we can plug this in for x in the first equation. We're taking out the x and substituting, or replacing, it with the 1.
Remember to write the solution as an ordered pair. The answer is the point that works for both equations. This solution is pretty easy to check by graphing. You can see in the graph below that the two lines cross at the point (1,3).
Here's a system that's a little bit harder: 2x + y = 11 and y = 3x - 9
It's not as simple as the first example because there's no easy x = some number that you can plug into the other equation. Let's pretend though that it had said y = 5 for the second equation. What would you do? You would plug in 5 for y into the first equation. Instead of just having y = some number, we have y = 3x - 9. No worries, we'll just plug the whole thing in! Take out the y in the first equation and replace it with 3x - 9:
A common mistake is for students to stop here and write x = 4 as the answer. Remember, the solution to a system is an ordered pair. We need both the x and the y to write the solution. We've just found half of the answer by finding that x = 4.
How do we find the y-coordinate? We can go back to the original system and plug in 4 for x into either equation. It does not matter which equation you choose - pick the one that looks easiest to you. We'll pick the second equation since that has a y by itself.
We could check our answer by graphing the two lines to see if they cross at the point (4,3). A faster way to check your answer is to go back to the other original equation (the one that you didn't use to find y) and plug in the values for x and y. If you use the same equation twice, it won't catch a mistake.
To sum it up, here are the steps that you can use when solving a system of equations with the substitution method:
Welcome to Kate's Math Lessons!
Teachers: make sure to check out the study guides and activities.