Multistep Equations

Multi-step Equations

Recall:

One and Two Step Equations

In our previous lesson, we looked at how to solve one and two step equations.
In order to solve an algebraic equation, we have to undo any operations to get the variable by itself.
An equation is like a balance or scale: Whatever we do on one side of the equation, we have to also do it on the other side.

Multi-step Equations

Multi-step equations involve more than two steps, but we solve them the same way we would any other equation.

Normally, there will be variables on both sides of the equation.

We may also have to distribute or perform additional operations.

Multi-step equations

Steps for solving:

Step 1: Move all variables (usually x) to one side

Step 2: Move all constants to the other side

Step 3: Undo any operations to get the variable by itself

Move -2 to the opposite side. We need to do +2 on both sides

We get: 2x = 16

Step 2:

Move 3x to the other side. We do this by subtracting 3x on both sides.

We get: 2x - 2 = 14

Step 1:

Example: 5x - 2 = 3x + 14

You can check your answer by plugging in your solution to each side of the equation and see if they are equal
(They should be!)

Optional:

Divide both sides by 2.

We get x = 7

Step 3:

Example: 5x - 2 = 3x + 14

Example 2: -5x - 6 = 2x + 8

Step 1: Move 2x over: -7x - 6 = 8

Step 2: Move -6 over: -7x = 14

Step 3: Divide by -7: x = -2

Optional: Check your answer

You Try!

Answer the following questions about solving multi-step equations!

Multiple Choice

4x + 10 = -26

• x = 4

• x = -4

• x = -9

• x = 9

Multiple Choice

Solve the equation for the variable given.
7m+8=71

• m = 10

• m = 9

• m = 12

• m = -10

Multiple Choice

5x - 4 + 2x = 3

• x = 1

• x = 2

• x = 3

• x = 4

Multiple Choice

Solve for x:
5x - 14 = 8x + 4

• x = 6
• x = -6
• x = -18/13
• x = 10/3

Multiple Choice

Solve for f:
f - 4 = 6f + 26

• f = -5
• f = 5
• f = 6
• f = -6

Multiple Choice

Solve the equation:
7x - 4 = 3 + 8x

• x = 7
• x = -7
• x = 2/7
• x = 11

Multiple Choice

14p - 8 = 22 + 20p

• p = -5

• p = 5

• p = 30

• p = -30

Multi-Step and Distribution

Distribution Review

Distribution is something we talked about when multiplying polynomials.
When we have a number outside of a set of brackets containing a binomial, trinomial, etc., that number gets multiplied by each term inside the brackets

For example: 2(x+4) = 2x + 8

Multi-Step and Distribution

When we solve a multi-step equation that has something that needs to be distributed, we have to do the distribution first before we can begin solving

Subtracting 21 on both sides, we get:

3x = -60

Step 2: Subtract 21

3(x + 7) = -39

3x + 21 = -39

Now it becomes a 2-step equation, which we solved last time

Step 1: Distribute

Example: 3(x+7) = -39

You can plug in your solution to the original equation to check your answer.

Optional:

3x = -60

Divide both sides by 3:

x = -20

Step 3: Divide by 3

Example: 3(x+7) = -39

Example 2: -2(x-3) = 10

Step 1: Distribute. -2x + 6 = 10

Step 2: Subtract 6. -2x = 4

Step 3: Divide by -2. x = -2

You Try!

Answer the following questions about solving multi-step distribution equations!

Multiple Choice

Solve the equation:

2(x + 3) = 8

• x = 2.5

• x = 1

• x = 7

• x5.5

Multiple Choice

Solve

-124 = 4 (1+ 8k)

• k = -11

• k = 10

• k = -4

• k = -2

Multiple Choice

-(n - 8) = -2

• n = 6

• n = 10

• n = -10

• n = -6

Multiple Choice

Solve the equation:
3(x - 1) = 2x + 9

• x = 12
• no solution
• x = 10
• x = 6

Multiple Choice

-11-5a = 6(5a + 4)

• a = -1

• a = -2

• a = -3

• a = -4

Multiple Choice

3(4r - 1) + 5 = 26

• r = -1

• r = 2

• r = -2

• r = 4

Going Forward

In Grade 9, we do not look at equations beyond these.

However, in future grades, you will be solving equations involving square roots, fractions (both with numbers and with variables), exponents, logarithms, etc.

Building these foundational skills will go a long way in being able to solve more complex equations in the future.