GED Mathematical Reasoning: Multi-step Equations

 

Steps for solving multi-step equations

  • First, simplify each side of the equal sign separately, if possible, by distributing to get rid of any parenthesis and combining like terms.
  • Second, add and subtract to get the variable terms on one side of the equation and the constant terms on the other side.
  • Third, multiply or divide to get the variable by itself on one side of the equal sign.
  • Finally, check your solution.

Example 1

Solve: 4x - 8 = 20

 

Focusing on the left side of the equal sign, where “x” is located, there are two operations: subtraction and multiplication (which is the operation between the 4 and the “x”).

So to isolate the variable “x” on the left side, we will need two steps involving opposite operations. We first need to add 8 to both sides. And then as a second step, we will divide both sides by 4.

When we add 8 to both sides, the subtraction and addition undo each other. This leaves just “4x” on the left side.

On the right side, we need to add 8 to 20, which is equal to 28.

Now we are left with a one-step equation: 4x equals 28

math 180

To solve for x, we will divide both sides by 4.

On the left side, 4 divided by 4 equals one, leaving just “x” on the left side.

On the right side, we need to divide 28 by 4, which is equal to 7.

So the solution to the equation is: x equals 7

algeb45

Let’s double check to be sure. When we substitute x = 7 back into the original equation, we get:  4(7) - 8 = 20 on the left side.

4(7) - 8 = 20
28 - 8 = 20
20 = 20

Since 20 equals 20 is a true statement, our solution checks out and we can be confident it is correct.

 

Example 2

Solve: 3x + 10 - 5x = 6

 

Looking at example two, you can see that this equation is a bit more involved. Let’s dive right in – following our four-step process.

First, we need to simplify each side of the equations separately, if possible. Although there are no parenthesis to get rid of, do you see that we can combine “like terms” on the left side? We can combine 3x and -5x, which is equal to -2x.

3x + 10 - 5x = 6
-2x + 10 = 6

Focusing on the left side, where “x” is located, we have two operations. Addition and the multiplication between the -2 and the “x”

Based on step two of the process, we will subtract 10 from both sides. This leaves just “-2x” on the left side.

On the right side, we subtract 10 from 6, which is equal to -4.

Now we are left with a one-step equation: -2x equals -4

math 181

Notice that we have successfully completed step two of the process – getting the variable terms on one side of the equal sign and the constant terms on the other.

Following step 3 of the process, we will now divide both sides by -2 to solve for x.

On the left side, -2 divided by -2 equals one, leaving just “x” on the left side.

On the right side, we need to divide -4 by -2, which is equal to positive 2.

algeb47

So the solution to the equation is: x equals 2

Let’s double check the solution to be sure it’s correct.

3x + 10 - 5x = 6
3(2) + 10 - 5(2) = 6
6 + 10 - 10 = 6

6 = 6

6 equals 6 is a true statement, so our solution checks out and we can be confident that it is correct.

 

Example 3

Solve: 9(a - 2) = 3a

 

The equation in example 3 contains parenthesis so let’s first distribute the left side.

9(a - 2) = 3a
9a - 18 = 3a

You may have already noticed that the variable “a” appears on both sides of the equal sign. So we’ll follow step two and add and subtract to get the variable terms on one side of the equation and the constant terms on the other side. There is more than one way to do this, but let’s begin by adding 18 to both sides.

When we add 18 to both sides, the addition and subtraction undo each other. This leaves just “9a” on the left side.

On the right side, we cannot add “3a” and 18 since they are not like terms, so we will leave it as: 3 “a” plus 18.

math 182

Next, we will subtract 3a from both sides. That will allow us to combine the variable terms, which are “like terms,” on the left side.

On the left side, 9a minus 3a is equal to 6a. On the right side, the 3a and -3a undo each other, leaving just 18.

algeb49We now have a one-step equation: 6a equals 18.

Following step 3 of the process, we will now divide both sides by 6 to solve for “a.”

On the left side, 6 divided by 6 equals one, leaving just “a” on the left side.

On the right side, we need to divide 18 by 6, which is equal to 3.

algeb50

So the solution to the equation is: a equals 3.

Finally, we’ll check our solution by substituting “a” equals 3 into the original equation.

9(a - 2) = 3a
9(3 - 2) = 3(3)
9(1) = 3(3)

9 = 9

Because 9 equals 9 is a true statement, our solution checks out and we can be confident that it is correct.

You have seen 1 out of 15 free pages this month.
Get unlimited access, over 1000 practice questions for just $29.99. Enroll Now