Algebra 1

Ch 3.3 – Solving Multi-Step Equations

Objective

Students will solve multi-step equations

Before we begin… In previous lessons we solved simple one-

step linear equations… In this lesson we will look at solving multi-

step equations…as the name suggests there are steps you must do before you can solve the equation…

To be successful here you have to be able to analyze the equation and decide what steps must be taken to solve the equation…

Multi-Step Equations This lesson will look at:

Solving linear equations with 2 operations Combining like terms first Using the distributive property Distributing a negative Multiplying by a reciprocal first

Your ability to be organized and lay out your problem will enable you to be successful with these concepts

Linear Equations with 2 Functions

Sometimes linear equations will have more than one operation

In this instance operations are defined as add, subtract, multiply or divide

The rule is First you add or subtract Second you multiply or divide.

Linear Equations with 2 Operations

You will have to analyze the equation first to see what it is saying…

Then, based upon the rules, undo each of the operations…

The best way to explain this is by looking at an example….

Example # 1

2x + 6 = 16

In this case the problem says 2 times a number plus 6 is equal to 16

Times means to multiply and plus means to add

The first step is to add or subtract.

To undo the addition of 6 I have to subtract 6 from both sides which looks like this:

2x + 6 = 16

- 6 -6

2x = 10

The 6’s on the left

side cancel

each other out

16 minus 6 equals

10

What is left is a one step equation 2x = 10

Example #1 (Continued) 2x = 10

To undo the multiplication here you would divide both sides by 2 which looks like this:

2x = 10

2 2The 2’s on

the left cancel out leaving x

x = 5

10 divided by 2 is

equal to 5

The solution to the equation 2x + 6 = 16 is x = 5

The second step is to multiply or divide

Combining Like Terms First When solving multi-step equations,

sometimes you have to combine like terms first.

The rule for combining like terms is that the terms must have the same variable and the same exponent.

Example: You can combine x + 5x to get 6x You cannot combine x + 2x2 because the

terms do not have the same exponent

Example # 27x – 3x – 8 = 24

I begin working on the left side of the equation

On the left side I notice that I have two like terms (7x, -3x) since the terms are alike I can combine them to get 4x.

7x – 3x – 8 = 24

4x – 8 = 24

After I combine the terms I have a 2-step equation. To solve this equation add/subtract 1st and then multiply/divide

Example # 2 (continued)

4x – 8 = 24

Step 1: Add/Subtract

Since this equation has – 8, I will add 8 to both sides

4x – 8 = 24

+8 +8

4x = 32The 8’s on the

left cancel

out

24 + 8 = 32

I am left with a 1-step equation

Example # 2 (continued)

4x = 32

Step 2: Multiply/Divide

In this instance 4x means 4 times x. To undo the multiplication divide both sides by 4

4x = 32

4 4The 4’s on

the left cancel out leaving x

x = 8

32 4 = 8

The solution that makes the statement true is x = 8

Solving equations using the Distributive Property

When solving equations, sometimes you will need to use the distributive property first.

At this level you are required to be able to recognize and know how to use the distributive property

Essentially, you multiply what’s on the outside of the parenthesis with EACH term on the inside of the parenthesis

Let’s see what that looks like…

Example #3

5x + 3(x +4) = 28

In this instance I begin on the left side of the equation

I recognize the distributive property as 3(x +4). I must simplify that before I can do anything else

5x + 3(x +4) = 28

+3x +125x = 28

After I do the distributive property I see that I have like terms (5x and 3x) I have to combine them to get 8x before I can solve this equation

Example # 3(continued)

8x + 12 = 28

I am now left with a 2-step equation

Step 1: Add/Subtract

The left side has +12. To undo the +12, I subtract 12 from both sides

8x + 12 = 28

-12 -12

8x = 16

The 12’s on the left cancel out leaving 8x

28 – 12 = 16

Example # 3 (continued)

8x = 16

Step 2: Multiply/Divide

On the left side 8x means 8 times x. To undo the multiplication I divide both sides by 8

8x = 16

8 8The 8’s on the left cancel out leaving x

16 8 = 2

The solution that makes the statement true is x = 2

x = 2

Distributing a Negative Distributing a negative number is

similar to using the distributive property.

However, students get this wrong because they forget to use the rules of integers

Quickly the rules are…when multiplying, if the signs are the same the answer is positive. If the signs are different the answer is negative

Example #4

4x – 3(x – 2) = 21

I begin by working on the left side of the equation.

In this problem I have to use the distributive property. However, the 3 in front of the parenthesis is a negative 3.

When multiplying here, multiply the -3 by both terms within the parenthesis. Use the rules of integers

4x – 3(x – 2) = 21

4x – 3x + 6 = 21

After doing the distributive property, I see that I can combine the 4x and the -3x to get 1x or x

Example # 4 (continued)

4x – 3x + 6 = 21

x = 15

After combining like terms you are left with a simple one step equation. To undo the +6 subtract 6 from both sides of the equation

x + 6 = 21

-6 -6

x + 6 = 21

The 6’s cancel

out leaving x

21 – 6 = 15

The solution is x = 15

Multiplying by a Reciprocal First Sometimes when doing the distributive

property involving fractions you can multiply by the reciprocal first.

Recall that the reciprocal is the inverse of the fraction and when multiplied their product is equal to 1.

The thing about using the reciprocal is that you have to multiply both sides of the equation by the reciprocal.

Let’s see what that looks like…

Example # 5

123

102 ( )x

In this example you could distribute the 3/10 to the x and the 2.

The quicker way to handle this is to use the reciprocal of 3/10 which is 10/3 and multiply both sides of the equation by 10/3

1210

3

10

3

3

102F

HGIKJFHG

IKJ ( )x

On the left side of the equation, after multiplying by the reciprocal 10/3 you are left with 120/3 which can be simplified to 40

On the right side of the equation the reciprocals cancel each other out leaving x + 2

The new equation is:

40 = x + 2

Example #5 (continued)

38 = x

After using the reciprocals you are left with a simple one-step equation

To solve this equation begin by working on the right side and subtract 2 from both sides of the equation

40 = x + 2

- 2 -2

40 = x + 2

40 – 2 = 38The 2’s on the right

side cancel out leaving

x

The solution to the equation is x = 38

Comments

On the next couple of slides are some practice problems…The answers are on the last slide…

Do the practice and then check your answers…If you do not get the same answer you must question what you did…go back and problem solve to find the error…

If you cannot find the error bring your work to me and I will help…

Your Turn

1. 6x – 4(9 –x) = 1062. 2x + 7 = 153. 6 = 14 – 2x4. 3(x – 2) = 185. 12(2 – x) = 6

Your Turn

9

23 27x b g

4

92 4 48xb g

6.

7.

8.

9.

10.

5m – (4m – 1) = -12

55x – 3(9x + 12) = -64

9x – 5(3x – 12) = 30

Your Turn Solutions

1. 14.2 2. 43. 44. 85. 1 ½

6. 37. -528. -139. -110. 5

Summary

A key tool in making learning effective is being able to summarize what you learned in a lesson in your own words…

In this lesson we talked about Solving Multi-step equations. Therefore, in your own words summarize this lesson…be sure to include key concepts that the lesson covered as well as any points that are still not clear to you…

I will give you credit for doing this lesson…please see the next slide…

Credit I will add 25 points as an assignment grade for you

working on this lesson… To receive the full 25 points you must do the following:

Have your name, date and period as well a lesson number as a heading.

Do each of the your turn problems showing all work Have a 1 paragraph summary of the lesson in your own

words Please be advised – I will not give any credit for work

submitted: Without a complete heading Without showing work for the your turn problems Without a summary in your own words…