This document provides instruction on solving one-step and two-step equations. It begins with the relevant common core standard and learning objectives. Useful vocabulary is defined, including equation, variable, inverse operations, and solution. The steps for solving one-step equations are outlined as looking at the operation on the variable, doing the inverse operation to both sides, and isolating the variable. Examples are worked through. Two-step equations are similarly explained as requiring simplification before inverse operations. More examples are given along with story problems to practice application. References are provided for images and vocabulary.

1. By: Justin Sahr
EDU 653
Solving Equations

2. 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?

3. Learning Objectives:
 I can…
 Solve one and two step equations
 Use an equation to model and solve a real-world situation

4. 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)

5. 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: CHECKYOURWORK.

6. 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

7. 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

8. 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

9. 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

10. 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 solvingANYTwo-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: CHECKYOURWORK.

11. 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

12. 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

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

14. 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.

15. 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 RoadTrip 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

16. 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

17. 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 RoadTrip Car“ wikimedia.org., 10 Jun.
2006.Web. 2 Jun. 2014.
 Wayzatarunning. “ConferenceXC“ wikimedia.org., 12 Oct. 2005.Web. 2 Jun.
2014.