Solving Equations

Solving algebra equations is difficult for many. If you struggle with algebra solving equations can be improved using this lesson and practice.

Multi-step equations

Let's look at another example:

4(2x + 3) = 68

First, we need to look to see if anything can be simplified. Remember in the previous section we talked about the number on the outside of the parentheses meaning multiplication? According to that, we can multiply 4 · 2x and 4 · 3. 4 · 2x is 8x and 4 · 3 is 12.

8x + 12 = 68

This gives us 8x + 12 = 68.

Now that both sides of the equal sign are simplified, we will need to use canceling to get x by itself. Right now we have two things we need to move, the 8 and the 12. We're adding 12, so we would subtract to move it. We're also multiplying x by 8, so we would divide to move it. But which one do we move first?

Remember, canceling uses inverse - or opposite - operations. Since we're using opposite operations to move things, we're going to use the opposite of the order of operations to decide which order to move them in.

The order of operations says we would simplify multiplication and division before addition and subtraction, so we're going to do the opposite. We're going to use addition/subtraction first, and then multiplication/division.

First, we'll subtract 12 from both sides:

8x+12=68
- 12-12

Since 12 - 12 is 0, we're left with 8x on the left. Since 68 - 12 is 56, we're left with 56 on the right.

8x=56
/ 8/ 8

Finally, we'll divide. 56/8 = 7

x=7

We're done! This means for 4(2x + 3) = 68, x has to equal 7.

Watch the video below to see this example problem solved.

Practice!

Let's practice what you just learned by going through a few more problems. Remember, in order to simplify these we'll use the order of operations and cancelling out.

Pay attention to the steps we take to simplify these expressions—in a bit, you'll have a chance to solve a few on your own.

Problem 1

Simplify this expression to find the value of x:

6x + 23 = 74

Take a moment to think about what you'd do first. You might even want to get out a piece of paper to see how you would simplify this on your own. Once you're ready, keep reading to see how we got the correct answer.

Just like we did on the last page, we'll start by seeing if there's anything we can do with the order of operations. This expression has two operations: addition and an exponent.

6x + 23 = 74

According to the order of operations, we need to calculate the exponent first. That's 23, which is equal to 2 ⋅ 2 ⋅ 2, or 8.

6x + 23 = 74

The order of operations says we should add next, but we can't add 6x + 8—a variable with a coefficient like 6x can only be added to another like term. (In other words, a number with the variable x can only be added to another number with the variable x.) In order to get 6x on its own, we'll have to cancel out + 8.

6x + 8 = 74

We can do that with the opposite of 8, which is - 8. We'll subtract 8 from both sides of the equals sign. 8 - 8 is 0. 74 - 8 is 66.

6x+ 8=74
- 8- 8

We're almost done. All that's left to do is get rid of the 6 in 6x. Remember, 6x is just another way of writing 6 ⋅ x.

6x = 66

Because 6 and x are being multiplied, we can cancel out the 6 by doing the opposite: dividing.

6x=66
66

6x / 6 is x and 66 / 6 is 11, so x = 11. We're done!

x = 11

As you might have noticed, you don't have to follow the order of operations once you start cancelling out. All that matters is keeping both sides of the expression equal. In fact, it's best to cancel out addition and subtraction first.

Problem 2

Let's try another problem. Simplify for y.

4 (3y - 8) = 4

This problem is slightly different from the last one, but it uses the same skills. Here's how to solve it:

According to the order of operations, we'll need to simplify the expression in parentheses first. However, we can't subtract 8 from 3y—we can't subtract a number from a variable.

4 (3y - 8) = 4

Because 4 is next to the parentheses, we're supposed to multiply what's in the parentheses by 4. (Confused? Review our lesson on reading algebraic expressions).

4 (3y -8) = 4

4 ⋅ 3y is 12y and 4 ⋅ -8 is -32. You can't subtract 32 from 12y, either, so to simplify this expression any further we'll have to start canceling things out.

12y - 32 = 4

Let's get rid of the -32 first. The opposite of -32 is 32, so we'll add 32 to both sides. -32 + 32 is 0, and 4 + 32 is 36.

12y- 32=4
+ 32+ 32

We're almost finished. We just have to cancel out of the 12 in 12y. Remember, 12y could also be written as 12 ⋅ y.

12y = 36

Because 12 and y are being multiplied, we can cancel out the 12 by dividing.

12y=36
1212

12y / 12 is y, and 36 / 12 is 3. We did it: y equals 3.

y = 3

Your turn

Try solving the next few problems on your own. The answers are below.

Problem 1

Simplify this expression to find the value of x:

-2 + x / 5 - 3 = 0

Problem 2

Find the the value of y:

3 (y + 2y) = 36

Problem 3

Find the the value of r:

300r - 60r + 102 = -380

Answers:

  1. x = 25
  2. y = 4
  3. r = -2