# How to Solve Multi-Step Equations

1. Math Lessons >
2. Solving Multi-Step Equations

### Solving Multi-Step Equations Step by Step

A multi-step equation is one that is solved by performing multiple steps. The steps might include performing the inverse operations, simplifying, and more. Let’s remember the steps in this order.

• Combine like terms if any and simplify each side of the equation.

• Isolate the variable on one side using inverse operations.

• Solve for the value of the variable.

### Solving Multi-Step Equations with Variables on One Side

Let’s take this equation with the variable x for instance.

–4 – 2x2 = 5

Here, there aren’t any like terms on either side. So let’s proceed with the inverse operations directly. As you can see, the left side is a fraction: (–4 – 2x) divided by 2. So, we’ll use its inverse, the multiplication to start with.

Multiplying both sides by 2, we have:

–4 – 2x2 . 2 = 5 . 2

Upon simplification, we’ll be left with:

–4 – 2x = 10

Note: Performing the same operation on both sides will make sure the equation remains balanced.

Our next step is to isolate the terms (–2x) on one side. So, we’ll add 4 to both sides and simplify.

–4 – 2x + 4 = 10 + 4

–2x = 14

Now, to isolate the variable by itself, we must divide both sides by –2 as the coefficient of x is –2.

–2x–2 = 14–2

⇨ x = –7

Thus, the solution of our equation is x = –7.

### Solving Multi-Step Equations with Variables on Both Sides

Let’s forge ahead, solving equations that involve the variables on both sides. Look at this equation.

–23 + 4z – 3 = –3z + 2

Let’s solve it step by step. We have combining like terms for our first step.

4z – 26 = –3z + 2

We have variables on both sides. So, we’ll bring the variable terms on the left and the constants on the right using these steps.

Adding 26 to both sides, we have:

4z – 26 + 26 = –3z + 2 + 26

4z = –3z + 28

Adding 3z to both sides, we have:

4z + 3z = –3z + 28 + 3z

7z = 28

As you can see, it’s become a lot easier to have the variable by itself on one side. All we need to do is divide both sides by 7, and we get

7z7 = 287

⇨ z = 4

Thus, the solution is z = 4.

Plug the value of the variable in the equation to check your answer.
Substituting z = 4 in –23 + 4z – 3 = –3z + 2, we get:
–23 + 4(4) – 3 = –3(4) + 2
–23 + 16 – 3 = –12 + 2
–10 = –10

### Solving Multi-Step Equations Using Distributive Property

What if there are parenthetical expressions in the equation? This is when we bring the distributive property into play. We expand the expression using the property and then press on with the usual combining-isolating-and-simplifying process. Let’s understand this with a few examples.

Example 1

Solve 5(3x + 12) = –4x + 3.

5(3x + 12) = –4x + 3

[The given equation]

5(3x) + 5(12) = –4x + 3

[Applying the distributive property on the left side]

15x + 60 = –4x + 3

[Simplifying]

15x + 60 – 60 = –4x + 3 – 60

[Subtracting 60 from both sides]

15x = –4x – 57

[Combining like terms]

15x + 4x = –4x – 57 + 4x

19x = –57

[Combining like terms]

19x19 = –5719

[Dividing both sides by 19]

⇨ x = –3

[Simplifying]

Example 2

Solve 4c – 7(1 + c) + 2(3c – 1) = –6c.

4c – 7(1 + c) + 2(3c – 1) = –6c

[The given equation]

4c – 7 – 7c + 6c – 2 = –6c

[Applying the distributive property on the left side]

3c – 9 = –6c

[Combining like terms]

3c – 9 + 9 = –6c + 9

3c = –6c + 9

[Simplifying]

3c + 6c = –6c + 9 + 6c

9c = 9

[Combining like terms]

9c9 = 99

[Dividing both sides by 9]

⇨ c = 1

[Simplifying]

Example 3

Solve 12(a – 2) = 713(a – 3).

12(a – 2) = 713(a – 3)

[The given equation]

1 . 13(a – 3) = 7 . 2(a – 2)

[Cross-multiplying]

13(a – 3) = 14(a – 2)

[Simplifying]

13(a) – 13(3) = 14(a) – 14(2)

[Applying the distributive property on both sides]

13a – 39 = 14a – 28

[Simplifying]

13a – 39 + 39 = 14a – 28 + 39

13a = 14a + 11

[Combining like terms]

13a – 14a = 14a + 11 – 14a

[Isolating the variable on one side]

–a = 11

[Combining like terms]

–a–1 = 11–1

[Dividing both sides by –1]

⇨ a = –11

[Simplifying]

### Key Takeaways

• The solution of a multi-step equation is obtained by performing multiple steps.

• To solve a multi-step equation with variable on one side, combine the like terms, perform the operations, and solve for the variable.

• To solve an equation with variables on both sides, perform inverse operations to isolate the variable on one side and solve for its value.

• To solve equations that involve parentheses, use the distributive property to get rid of the parentheses and proceed with the usual three-step process. Upshift your practice with our free printable Multi-Step Equations worksheets!

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