Systems of Linear Equations

Systems are two different equations that share a common point. There are three methods to finding a the point that works for both equations. The first method is called graphing. In my previous post I explained this method and how to use it (link here). The graphing method is more visual, but requires you to draw the line which isn’t always possible. The two other methods are more algebraic, and don’t require graphing paper. The first is substitution and the second is elimination. I will be explaining substitution in this post.

Substitution

This method works by isolating equations then mixing them. This method may appear complicated, but it works with most formats of equations, all you have to do is rearrange them. Don’t forget to verify your results.

The equations we are going to use:
6x + 3y = 9
9x – y = 8

Step 1: Isolate one of the variables (x or y) from one of the above equations. Isolating means you want one of the variables by it’s self on one side of the equation. We are going to isolate y in the second equation. It’s also important to remember that picking the equation with smaller numbers, and no fractions or negatives will be easier. I am picking the one with the negatives to show how you would do this if you had negatives.

9x – y = 8
9x = 8 + y (move the – y to the other side)
9x – 8 = y (move the variable to the other side)

Step 2: Now that you know what y equals you can now mix equations. Insert the x = … or y = … into the other equation. Fun Fact: When you mix equations you should get a vertical or horizontal line!

6x + 3y = 9
6x + 3(9x – 8) = 9 (mix equations)
6x + 27x – 24 = 9 (distribute the 3)
33x – 24 = 9 (combine like terms)
33x = 33 (add 24 to both sides of equation)
x = 1 (divide by 33 on both sides)
solution: x = 1

Step 3: Place the solution for x in the place of the x variable in the equation to solve for y (or the variable you didn’t use).

9x – 8 = y
9 (1) – 8 = y
9 – 8 = y
1 = y

Step 4: Write out your system as a coordinate.

(1,1)

Step 5 (optional, but recommended): Verify your answer by replacing the x and y variables in the original equation and determining if they are true.

6x + 3y = 9
6 (1) + 3 (1) = 9
6 + 3 = 9
9 = 9 (true)

9x – y = 8
9(1) – 1 = 8
9 – 1 = 8
8 = 8 (true)

**Tip: If you want to verify visually one way to do this is by using Desmos, but most likely your teacher won’t let you use Desmos during a test. Desmos is a free graphing online calculator.**

With our equation the first equation is in red and the second equation is in blue.

Final Reflection on Math 10

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