By: Justin Sahr

EDU 653

Solving Equations

Common Core State Standard

CCSS.MATH.CONTENT.7.EE.B.4.A

Solve word problems leading to equations of the

form px + q = r and p(x + q) = r, where p, q, and r are

specific rational numbers. Solve equations of these forms

fluently. Compare an algebraic solution to an arithmetic

solution, identifying the sequence of the operations used in

each approach. For example, the perimeter of a rectangle is 54 cm.

Its length is 6 cm. What is its width?

Learning Objectives:

I can…

Solve one and two step equations

Use an equation to model and solve a real-world situation

Useful Vocabulary Equation: A mathematical statement that says that two

expressions have the same value; any number sentence with an =

sign.

Variable: A letter used to represent a number value in an

expression or an equation.

Inverse Operations: Two operations that have the opposite

effect, such as addition and subtraction.

Solution: The value of a variable that makes an equation true. **Taken from Math.com on 2 Jun, 2014**

SADMEP: Following the reverse order of operations to solve an

equation (Subtract, Add, Divide, Multiply, Exponents, Parenthesis)

Solving One-Step Equations

Why are these called one-step equations?

There is only one thing you need to do to solve it!

Here are the steps to solving ANY One-Step Equations:

STEP 1: Look to see what is being done to the variable and do

the opposite. (We call this the inverse operation)

STEP 2: What you do to one side you must do to the other

side. (Other side of the = sign)

STEP 3: Isolate the variable. (What does it mean to isolate

someone or something?)

STEP 4: CHECK YOUR WORK.

Let’s try:

EX 1:

STEP 1: What is being done?

adding 5

What is the inverse operation?

subtracting 5

STEP 2: Do the same to both sides.

STEP 3: Isolate the variable.

STEP 4: Check your work.

– 5 – 5 x + 0 = 7

So, x = 7

x + 5 = 12

(7) + 5 = 12

x + 5 = 12

Let’s try: EX 2:

STEP 1: What is being done?

dividing by 2

What is the inverse operation?

multiplying by 2

STEP 2: Do the same to both sides.

STEP 3: Isolate the variable.

STEP 4: Check your work.

2* *2

The twos cancel out

So, x = 28

𝑥

2= 13

2𝑥

2= 28

28

2= 14

𝑥

2= 13

Whiteboard Practice:

Take out you whiteboards and practice these problems using

the steps we just went over. Once you have been approved by

myself you may move on to the next problem.

1. 2.

3. 4.

𝑥

3= −6 x - 5 = -9

-3x = 12

4x = -24

Story Problem

If you were to get paid $15 for every lawn you mow. Your

goal is $165 for a new bike. How many lawns do you need to

mow?

Halley from Boston. “Mowing the Lawn-Half-Cut“

wikimedia.org., 20 May. 2005. Web. 2 Jun. 2014.

• First let’s look for important information:

• $15 for every lawn

• “for every” means multiply

• “is” usually means =

• $165 is our goal

• “how many lawns” is what we do not

know, or our variable.

• Write the equation from what you

have underlined:

15

15x = 165

x = 11 lawns

Solving Two-Step Equations Why are these called two-step equations?

There is more than one thing you need to do to solve it!

Here are the steps to solving ANY Two-Step Equations:

STEP 1: Simplify (combine like terms and/or distribute)

STEP 1: Look to see what is being done to the variable and do the opposite. (We call this the inverse operation) Since more than one thing is being done, you must chose each step following

the reverse order of operations.

STEP 2: What you do to one side you must do to the other side. (Other side of the = sign)

STEP 3: Isolate the variable. (What does it mean to isolate someone or something?)

STEP 4: CHECK YOUR WORK.

Let’s try: EX 3:

STEP 1: What is being done?

adding 3 and multiplying by 2

What is the inverse operation of each?

subtracting 5 and dividing by 2

What order do we go in (SADMEP)?

subtract then divide

STEP 2: Do the same to both sides.

STEP 3: Isolate the variable.

STEP 4: Check your work.

– 3 – 3 2x + 0 = -20

So, 2x = -20

2x + 3 = -17

x = -10

2x + 3 = -17

2 2

2(-10) + 3 = -17

-20 + 3 = -17

-17 = -17

Let’s try: EX 4:

STEP 1: What is being done?

subtracting 5 and dividing by -2

What is the inverse operation of each?

adding 5 and multiplying by 2

What order do we go in (SADMEP)?

add then multiply

STEP 2: Do the same to both sides.

STEP 3: Isolate the variable.

STEP 4: Check your work.

+ 5 + 5

(-2) (-2)

x = -36

𝑥

−2− 5 = 13

𝑥

−2− 5 = 13

𝑥

−2= 18

(−36)

−2− 5 = 13

18 – 5 = 13

13 = 13

Whiteboard Practice:

Take out you whiteboards and practice these problems using

the steps we just went over. Once you have been approved by

myself you may move on to the next problem.

1. 2.

3. 4.

𝑥

3− 3 = −6 -5x - 5 = -25

-3x + 2 = 20

5 - 4x = -19

Whiteboard Practice:

1. 2.

3. 4.

𝑥

3− 3 = −6 + 9 -2x + 5(x + 2)= 25

-2(x + 2) – 4 = 20

5x - 4x = -19

Here are a few examples of simplifying before you can solve.

Story Problem

You and some friends are going on a road trip. You have gone

60 miles, and you are driving at a rate of 35 mph. How long

will it take you to travel the 200 mile trip?

Nigaard, Stig. “Erik’s Air Conditioned Road Trip Car“

wikimedia.org., 10 Jun. 2006. Web. 2 Jun. 2014.

• First let’s look for important information:

• 60 miles so far

• 35 miles per hour

• “per” means multiply

• “how long” (hours) is what we do

not know, or our variable.

• 200 miles is our goal

• Write the equation from what you

have underlined:

60 + 35x = 200

35x = 140

x = 4 hours

Story Problem

Mike and Chris are running a marathon. Mike is 24 meters

ahead of Chris, but he is losing ground at a rate of 8 meters

per hour. How long will it take mike to still be a

comfortable 4 meters ahead of Chris?

Wayzatarunning. “ConferenceXC“

wikimedia.org., 12 Oct. 2005. Web. 2 Jun. 2014.

• First let’s look for important information:

• 24 meters ahead

• “losing” means subtracting

• 8 meters per hour

• “per” means multiply

• “how long” (hours) is what we do

not know, or our variable.

• 4 meters ahead is our goal

• Write the equation from what you

have underlined:

24 – 8x = 4

-8x = -20

x = 2.5 hours

References

Some vocabulary taken from math.com

Photos Halley from Boston. “Mowing the Lawn-Half-Cut“ wikimedia.org., 20 May.

2005. Web. 2 Jun. 2014.

Nigaard, Stig. “Erik’s Air Conditioned Road Trip Car“ wikimedia.org., 10 Jun.

2006. Web. 2 Jun. 2014.

Wayzatarunning. “ConferenceXC“ wikimedia.org., 12 Oct. 2005. Web. 2 Jun.

2014.