Multi-step equations are simply equations that require more than 2 steps to solve for the variable. The same tools that can be used to solve more simple equations – including the order of operations and the Properties of Equality – can also be applied here. However, there is another important step that must be performed first: combining like terms. Additionally, in some cases, we need to apply the Distributive Property before we can even begin combining like terms.

Like terms have the same variable(s) raised to the same power(s). For example, 2y and -9y are like terms, 7 and 8 are like terms (whatever variable they have is raised to the 0th power for both), and g² and 2g² are like terms; however, 4y and 4x are not like terms, as aren’t 4y and 4y². In the second case, y and y² are raised to different powers and thus cannot be combined.

With that little introduction, let’s see how we can solve some multi-step equations.

Suppose we’re given this problem.

Solve –x + 9(x – 1) + 3 = 42 + 2x for x.

Where do we start?

First, since we can see the form 9(x – 1) in the expression, we can immediately apply the Distributive Property. This results in:

–x + 9(x – 1) + 3 = 42 + 2x

–x + 9x – 9 + 3 = 42 + 2x

Now, our next step is to combine like terms. There are no like terms in the right-hand side, but there are in the left-hand side, so we can simplify there:

–x + 9x – 9 + 3 = 42 + 2x

(-x + 9x) + (-9 + 3) = 42 + 2x 

8x + (-6) = 42 + 2x

8x – 6 = 42 + 2x

Now that we’ve combined like terms, we can go ahead and use the Properties of Equality to solve for x. It is helpful to group all x-terms on one side of the equation and all constants (numbers without variables) on the other side:

8x – 6 = 42 + 2x 

8x – 2x – 6 = 42

8x – 2x = 42 + 6

Now, we can combine like terms again:

8x – 2x = 42 + 6

6x = 48

Finally, to solve for x, we divide both sides by 6:

6x = 48

x = 8

We can check our answer to ensure we got it right:

For x = 8

-(8) + 9(8 – 1) + 3 = 42 + 2(8)

-8 + 9(7) + 3 = 42 + 16

-8 + 63 + 3 = 42 + 16

55 + 3 = 42 + 16

58 = 58 ✔

Let’s do another problem.

Solve 6/x + 5 = (2 + x)/x.

How do we solve this problem?

We can apply many of the same tools we used in the last problem, but in a different order.

First, we need to get the x out of the denominators. We can do this by using the Multiplication Property of Equality to multiply both sides of the equation by x. This gives us:

6/x + 5 = (2 + x)/x

6x/x + 5x = (2 + x)x/x

6 + 5x = 2 + x

Now, we group the variables on one side of the equation and the constants on the other.

6 + 5x = 2 + x

6 + 4x = 2

4x = -4

Finally, we isolate x by dividing both sides of the equation by 4:

4x = -4

x = -1

Now, we check our answer.

For x = -1

6/x + 5 = (2 + x)/x 

6/(-1) + 5 = (2 + (-1))/(-1)

-6 + 5 = (2 – 1)/(-1)

-6 + 5 = 1/(-1)

-6 + 5 = -1

-1 = -1 ✔

A key takeaway from this post is that not all multi-step equations are made equal. Sometimes, you need to be a bit creative with how you apply the main principles to solve the equation. Often, however, you can just use reason and intuition to accomplish this.

Summary

In summary, there are general tools to help you solve the equations. In many cases, you can follow this order:

1. Apply the Distributive Property.
2. Combine like terms.
3. Apply the Properties of Equality.
4. Simplify and combine like terms.
5. Solve for the variable.

The general strategy you would follow is:

1. Simplify the left-hand side and right-hand sides.
2. Group all variable terms on one side and all constants on the other side of the equation.
3. Solve for the variable.

If you have any questions or would like some help with problems like these, feel free to send me a message. 

Thank you for reading this post, and I know you’ll ace solving multi-step equations 🙂

~ Your Humble Study Buddy, 

Ace