This document provides a lesson on solving equations by addition and subtraction. It begins by introducing key terms like variables, expressions, and equations. It then shows how to solve equations using the addition and subtraction properties of equality. Examples are provided of solving equations step-by-step and translating word problems into mathematical expressions. The document concludes with practice problems for students to assess their understanding of solving equations by addition and subtraction.

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LESSON 4 – SOLVING EQUATIONS BY ADDITION AND
SUBTRACTION
One of the most important qualities of mathematics is that it is a science of patterns.
LessonObjectives
Solve an equation for a variable.
Translate verbal phrases into mathematical expressions.
Solve applications including variables, with a problem-solving process
The key terms listed below will help you keep track of important mathematical words and
phrases that are part of this lesson. Look for these words and circle or highlight them
along with their definition or explanation as you work through the MiniLesson.
 Variable
 Algebra
 Algebraic Expression or Expression
 Algebraic Equations or Equation
 Addition Property of Equality
 Inverse Operations
 Solution
KEY TERMS
INTRODUCTION

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MINILESSON
Communicating about patterns is often made easier by using a letter to represent all of the
numbers fitting a pattern. We call such a letter a variable. A variable is a symbol (usually a
letter) that stands for a value that may vary. For example, in an earlier lesson we learned
about the addition property of zero, which states that the sum of 0 and any number is that
number. We can demonstrate this property as
0 + 1 = 1
0 + 2 = 2
0 + 3 = 3
0 + 4 = 4
0 + 5 = 5
… the pattern continues on indefinitely. All whole numbers fit this pattern. We can
communicate this pattern for all whole numbers by letting a number, such as a, represent all
whole numbers. Now we can write this pattern as
0 + a = a
Using variable notation is a primary goal of learning algebra. We now take some important
first steps in beginning to use variable notation.
When we combine numbers and variables using operations (such as addition, subtraction, and
others we have not yet covered) the resulting combination of mathematical symbols is called
an algebraic expression, or an expression.
a + 2 and x - 4
are examples of expressions
When we have a mathematical statement where two expressions that are equal, this is called
an algebraic equation, or an equation. The key difference between an expression and an
equation is the presence of an equal sign. So, for example
a + 2 = 5 and x – 4 = 6
are examples of equations.
VARIABLES
MATHEMATICAL EXPRESSIONS
MATHEMATICAL EQUATIONS

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To solve an equation, we use properties of equality to write simpler equations, all equivalent
to the original equation, until the final equation has the form
x = number or number = x
Equivalent equations have the same solutions, so the word “number” above represents the
solution of the original equation. The first property of equality that helps us to write simpler,
equivalent equations is the addition property of equality.
To simplify these ideas, think of them this way. The same number may be added or subtracted
from both sides of the equation without changing the solution of the equation. A good way
to visualize a true equation is to picture a balanced scale. Since it is balanced, each side of
the scale weighs the same amount. Similarly, in a true equation the expressions on each side
have the same value. Picturing our balanced scaled, if we add weight to each side, the scale
remains balanced.
SOLVING EQUATIONS BY ADDITION AND SUBTRACTION
AdditionProperty of Equality
States that if you add a number to both sides of an equation, the sides remain equal
Let a, b and c represent numbers. Then
a = b
and a + c = b + c
are equivalent equations
Subtraction Property of Equality
States that if both sides of an equation have the same number subtracted from them, the
sides remain equal
Let a, b and c represent numbers. Then
a = b
and a – c = b - c
are equivalent equations

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Example 1: Solve x - 3 = 7
Step 1 To solve the equation for x,we need to rewrite the equation in the form
x = number. In other words, our goal is to get x alone on one side of the
equation. To do so, we add 3 to both sides of the equation.
x – 3 + 3 = 7 + 3 add 3 to both sides of the equation
Step 2 Complete the expressions on each side of the equation.
x = 10 – 3 + 3 = 0 and 7 + 3 = 10
Check To check your answer, replace x with 10 in the original equation.
x – 3 = 7 original equation
10 – 3 = 7 replace x with 10
7 = 7 True
Since 10 = 10 is a true statement, 10 is the solution of the equation.
Additionand Subtraction as Inverse Operations
Two important observations:
The inverse of addition is subtraction. If we start with a number z and add a number
y, then subtracting y from the result will return us to the original number z.
z + y – y = z
The inverse of subtraction is addition. If we start with a number z and subtract a
number y, then adding y from the result will return us to the original number z.
z - y + y = z

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Example 2: Solve x + 5 = 12
Step 1 Subtract 5 from both sides of the equation.
x + 5 - 5 = 12 - 5
Step 2 Complete the expressions on each side of the equation.
x = 7 5 - 5 = 0 and 12 – 5 =7
Check To check your answer, replace x with 7 in the original equation.
x + 5 = 12 original equation
7 + 5 = 12 replace x with 7
12 = 12 True
Since 12 = 12 is a true statement, 7 is the solution of the equation.
Example 3: Solve 8 = a - 1
Step 1 8 + 1 = a -1 + 1
Step 2 9 = a
Check 8 = a - 1 original equation
8 = 9 - 1 replace a with 9
8 = 8 True
YOU TRY
1. Solve x – 8 = 10 for x.
2. Solve 11 = y + 5 for y.
3. Show that 3 is a solution of the equation a + 2 = 5

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There are a wide variety of phrases that translate into mathematical expressions. Some
examples are shown in this table, though the list is far from complete.
Phrase Translates to: Phrase Translates to:
sum of x and 12 x + 12 difference of x and 12 x - 12
9 greater than b b + 9 9 less than b b - 9
6 more than y y + 6 6 subtracted from y y- 6
21 plus z 21 + z 21 minus z 21 - z
3 larger than n n + 3 3 smaller than n n - 3
Phrases are sums Phrases are differences
Example 4: Translate the following phrases into mathematical expressions:
(a) “8 larger than n”
(b) “12 less than z”
(c) “x decreased by 12”
Solution. Here are the translations.
(a) “8 larger than n” becomes n + 8
(b) “12 less than z” becomes z - 12
(c) “x decreased by 12” becomes x - 12
TRANSLATING WORDS INTO MATHEMATICAL EXPRESSIONS
YOU TRY
4. Translate
a. “the sum of 7 and k”
b. “18 subtracted from q”
c. “4 smaller than x plus 6”
d. “the sum of 15 and 8 less than a”

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Example 5: Let W represent the width of a rectangle. The length of a rectangle is 4 feet longer
than its width. Express the length of the rectangle in terms of its width W.
Step 1 We know that the width of the rectangle is W. Because the length of the
rectangle is 4 feet longer than the width, we must add 4 to the width to find the
length.
Length is 4 more than the width
Length = 4 + W
Solution The length of the rectangle, in terms of its width W, is W + 4
GIVEN: [Write down the information that is provided in the problem. Diagrams can be
helpful as well.]
GOAL: [Write down what it is you are asked to find. This helps focus your efforts.]
MATH WORK: [Show your math work to set up and solve the problem.]
CHECK: [Is your answer reasonable? Does it seem to fit the problem? A check may not
always be appropriate mathematically but you should always look to see if your result
makes sense in terms of the goal.]
FINAL RESULT AS A COMPLETE SENTENCE: [Address the GOAL using a
complete sentence.]
APPLICATIONS WITH EQUATIONS

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YOU TRY
5. A string measuring 15 inches is cut into two pieces. Let s represent the length of
one of the resulting pieces. Express the length of the second piece in terms of the length
s of the first piece.

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LESSON 4 - PRACTICE PROBLEMS
1. Is 23 a solution to the equation 4 = y – 11?
2. Are the equations x + 2 = 9 and x = 7 equivalent?
3. Solve k – 9 = 8 for k.
4. Solve z + 4 = 6 for z.
5. Solve 12 = y + 5 for y.
6. Four more than a certain number is 12. Find the number.
7. A certain number reduced by six is 8. Find the number.

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8. Amelie withdraws $125 from her savings account. Because of the withdrawal, the current
balance in her account is now $645. What was the original balance in the account before
the withdrawal?
9. The perimeter of a triangle is 221 feet. Two of the sides of the triangle measure 70 feet and
50 feet, respectively. Find the measure of the third side of the triangle.
10. In 2010, Cuyahoga Community College obtained an additional 106 acres of land adjacent
to the Eastern campus. The land increase the total acreage of the campus to 218 acres.
What was the acreage of the campus prior to obtaining this additional land?

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LESSON 4 – ASSESS YOUR LEARNING
Work the following to assess your learning of the concepts in this lesson. Try to write
complete solutions and show as much work as you can.
1) Solve 18 – x = 7 for x.
2) Solve 4 = 13 – n for n.
3) Solve 45 = 17 + k for k.
4) Solve 20 = 3 – x for x.
5) Solve x + 7 = 22 for x.
6) 9 greater than a number is 11. Find the number.
7) 13 smaller than a number is 2. Find the number.

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8) The difference between a number and 17 is 24. Find the number.
9) A number subtracted from 89 is 43. Find the number.
10) A triangle has a perimeter of 55 feet. Side a = 20 feet. Side b is 3 feet less than side a. How
long is side c?
11) Burt makes a deposit in his account that brings his balance up to $1,324. Prior to the deposit
his balance was $987. How much was the deposit for?

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12) Gary has two quadcopters that he built. The Hawk has a battery life of 215 minutes. The
Eagle has a battery life of 2 hours and 20 minutes longer than the Hawk. How long does the
battery of the Eagle last?
13) Mount Hood, in Oregon, has an elevation of 11,239 feet. The peak of Mt. Everest is x feet
higher in elevation than the peak of Mount Hood. The peak of Kilimanjaro is y feet higher
lower elevation than Mt. Everest.
i) Write an expression representing the height of Mt. Everest.
ii) Write an expression representing the height of Mr. Kilimanjaro.
iii) If Denali has an elevation of 20,322 feet, which is 8,707 feet less than Mt. Everest. What
is the height of Mt. Kilimanjaro?

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14) A rectangle has a width that is 5 cm longer than the height.
i) Write an expression representing the height.
ii) Write an equation representing the perimeter.