Download One-Step Equations and Inverse Operations

One-Step Equations and
Inverse Operations
Andrew Gloag
Melissa Kramer
Anne Gloag
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AUTHORS
Andrew Gloag
Melissa Kramer
Anne Gloag
EDITOR
Annamaria Farbizio
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Printed: March 11, 2013
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C ONCEPT
Concept 1. One-Step Equations and Inverse Operations
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One-Step Equations and
Inverse Operations
Here you’ll learn how to isolate variables using inverse operations in order to solve equations in one step.
What if you were in a math contest and were given the equation x + 4 = 16? Or how about the equation 9x = 72?
Could you solve each of these equations in one step? After completing this Concept, you’ll be able to solve equations
like these in a just a single step by using inverse operations.
Guidance
It’s Easier than You Think
You have been solving equations since the beginning of this textbook, although you may not have recognized it. For
example, in a previous Concept, you determined the answer to the pizza problem below.
$20.00 was one-quarter of the money spent on pizza.
1
4m
= 20.00 What divided by 4 equals 20.00?
The solution is 80. So, the amount of money spent on pizza was $80.00.
By working through this question mentally, you were applying mathematical rules and solving for the variable m.
Definition: To solve an equation means to write an equivalent equation that has the variable by itself on one side.
This is also known as isolating the variable.
In order to begin solving equations, you must understand three basic concepts of algebra: inverse operations,
equivalent equations, and the Addition Property of Equality.
Inverse Operations and Equivalent Equations
In another Concept, you learned how to simplify an expression using the Order of Operations: Parentheses, Exponents,
Multiplication and Division completed in order from left to right, and Addition and Subtraction (also completed from
left to right). Each of these operations has an inverse. Inverse operations “undo” each other when combined.
For example, the inverse of addition is subtraction. The inverse of an exponent is a root.
Definition: Equivalent equations are two or more equations having the same solution.
The Addition Property of Equality
Just like Spanish, chemistry, or even music, mathematics has a set of rules you must follow in order to be successful.
These rules are called properties, theorems, or axioms. They have been proven or agreed upon years ago, so you can
apply them to many different situations.
For example, the Addition Property of Equality allows you to apply the same operation to each side of an equation,
or “what you do to one side of an equation you can do to the other.”
The Addition Property of Equality
For all real numbers a, b, and c:
If a = b, then a + c = b + c.
Solving One-Step Equations Using Addition or Subtraction
Because subtraction can be considered “adding a negative,” the Addition Property of Equality also works if you need
to subtract the same value from each side of an equation.
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Example A
Solve for y :
16 = y − 11.
Solution: When asked to solve for y, your goal is to write an equivalent equation with the variable y isolated on one
side.
Write the original equation: 16 = y − 11.
Apply the Addition Property of Equality: 16 + 11 = y − 11 + 11.
Simplify by adding like terms: 27 = y.
The solution is y = 27.
Example B
Solve for z :
5 = z + 12
Solution:
Apply the Addition Property of Equality:
5 = z + 12
5 − 12 = z + 12 − 12
5 − 12 = z
−7 = z
The solution is −7 = z.
Equations that take one step to isolate the variable are called one-step equations. Such equations can also involve
multiplication or division.
Solving One-Step Equations Using Multiplication or Division
The Multiplication Property of Equality
For all real numbers a, b, and c:
If a = b, then a(c) = b(c).
Example C
Solve for k : −8k = −96.
Solution: Because −8k = −8 × k, the inverse operation of multiplication is division. Therefore, we must cancel
multiplication by applying the Multiplication Property of Equality.
Write the original equation: −8k = −96.
Apply the Multiplication Property of Equality: −8k ÷ −8 = −96 ÷ −8.
The solution is k = 12.
When working with fractions, you must remember:
division, you really must multiply by its reciprocal.
2
a
b
× ba = 1. In other words, in order to cancel a fraction using
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Concept 1. One-Step Equations and Inverse Operations
Guided Practice
1. Determine the inverse of division.
2. Solve 18 · x = 1.5.
Solutions:
1. To undo division by a number, you would multiply by the same number.
2. The variable x is being multiplied by one-eighth. Instead of dividing two fractions, we multiply by the reciprocal
of 81 , which is 8.
1
8
· x = 8(1.5)
8
x = 12
Practice
Sample explanations for some of the practice exercises below are available by viewing the following video. Note
that there is not always a match between the number of the practice exercise in the video and the number of the
practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba
sic Algebra:One-Step Equations (12:30)
MEDIA
Click image to the left for more content.
Solve for the given variable.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
x + 11 = 7
x − 1.1 = 3.2
7x = 21
4x = 1
2
5x
12 = 3
x + 52 = 32
x − 56 = 83
0.01x = 11
q − 13 = −13
z + 1.1 = 3.0001
21s = 3
t + 12 = 31
7f
7
11 = 11
3
1
4 = −2 ·y
3
6r = 8
9b
3
16 = 8
3