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Section 2.1
Equations
Pre-Activity
Preparation
Can you figure it out?
• You have $1000 to invest at a 5.2% interest rate for 3 years.
How do you calculate interest earned on the account?
• If you have 550 miles to travel and your average speed is
70 mph, how do you figure out how many hours it will take to
get there?
• If a map is in proportion of two inches to one mile, can you find
out how many miles correspond to 5 inches? Can the average
person walk that far or does he or she need to drive?
• If you know dinner cost $26.95 per person (gratuity or tip included), can you find out how much a
dinner party for 125 would cost? What percentage is that of a wedding budget of $10,000?
• If it takes 23 square yards of material to construct a tarp for a small boat, how many square yards
would it take to make 12 tarps? What percentage of a bolt of 500 square yards would the tarps use?
How many more tarps could be made?
In each case, an equation makes it possible to efficiently and systematically answer the question posed.
Learning Objectives
• Understand and use the properties of equations
• Solve proportion equations
• Solve percent equations
• Apply equation properties to solve equations
Terminology
Previously Used
New Terms
to
Learn
coefficient
amount
proportion
constant
base
proportion equation
factor
equal
similar triangles
ratio
equation
solution
reciprocal
equivalent
solve
substitute
formula
unknown
variable
percent
93
Chapter 2 — Solving Equations
94
Building Mathematical Language
Equations
• Two equal quantitites
Operation sign
An equation is a mathematical statement that two quantities
are the same: the left and right sides of the equation are
equivalent. Sometimes these statements are true; sometimes
they are false.
Variable
Equal sign
3 x − 7 = 11
• A solution
Constants
To solve an equation containing a variable (an unknown)
is to find the value of the variable that makes the equation true. In the equation 3x − 7 = 11, if the value
of the variable is 6, the statement is true since 18 − 7 = 11. If the value of the variable is 8, the statement
is false since 24 − 7 ≠ 11. The value 6 is, therefore, a solution to the equation; the value of 8 is not.
Types of Algebraic Equations
Proportion
equations
Proportions are equations made up of two equal or equivalent ratios:
a c
=
b d
There are four positions for a possible unknown quantity:
x 5
=
24 7
Solving
Application
Percent
equations
3
x
=
5.7 21
5 38
=
x 70
1.5 3.33
=
8
x
If three of the quantities are known, solve for the unknown quantity by following
the Methodology for Solving Proportional Equations.
Similar triangles have the same
a
b
shape and same size angles, but not
necessarily the same length sides. The
c
ratios of corresponding sides of similar triangles are equal.
d
e
f
a b c
= =
d e f
Unit conversions depend on the use of proportion equations: if you know that 1 foot = 12 inches, you can convert
any measurement in feet to inches, and visa versa.
1 foot
16 feet
=
12 inches x inches
Recipes are still another application because you can
use proportions to decide how to make the same basic
recipe yield additional servings.
2 cups
x cups
=
8 servings 75 servings
Recall that P percent is the number
P
25
. For example, 25% is
or 0.25 and
100
100
P is the percent number. Percent equations have direct application to many real
world situations, such as some of the examples given in the introduction to this
activity. The basic percent equation is: P(percent) × B(base) = A(amount)
For example, 25% × 120 = 30 or 0.25 × 120 = 30. Here, 25% is the percent, 120 is
the base, and 30 is the amount.
Section 2.1 — Equations
Percent
equations
(continued)
95
There are three types of percent problems:
Type 1 The amount is unknown:
Type 2 The percent is unknown:
Type 3 The base is unknown:
What is 15% of 27.50?
7.85 is what percent of 35.7?
93 is 18% of what?
Use your translation skills from Section 1.5: is means “=”, of identifies the base, and
what is the quantity you are trying to determine.
Solving
Mathematically, there are three quantities; if two quantities are known, then the other
quantity can be found by applying the Methodology for Solving Percent Equations.
Alternate method: percent equations are the same as proportion equations when the
percent is written as a ratio:
P
and the ration of the amount to base as the second. The
100
P
A
= . You may choose to
resulting proportion is known as the percent proportion:
100 B
Write P% as a the first ratio
use the percent equation or the percent proportion to solve percent problems.
Applications
The calculation of interest is a very common application of percent equations.
Survey or election results are very often reported as percents: Of the 2000 people
surveyed about public transportation, 35% agreed that ...
Sales tax is figured as a percent of a purchase: If the sales price of a new computer is
$500 and the sales tax rate is 7.25%, what is the total amount you’ll be paying?
Formulas
Solving
Formulas, which are also called literal equations in some textbooks, are equations
that are made up of numeric relationships or measurements that express a fact, like
the area of a circle or the distance a car travels. Formulas may use several variables.
Usually there is only one unknown variable in a problem and all other variable
values (or measurements) will be given.
Each variable can be solved in terms of the other variables so that one relationship
can be expressed by several different formulas.
Basic equation: d = rt
For example, the formula for finding distance
D
traveled over time (Distance = Rate × Time or
Solve for rate: r =
t
d = rt) can also be used for finding the rate of travel
D
(r) or the length of time it took to travel (t):
Solve for time: t =
r
Section 2.3 will go into much greater detail about manipulating and solving formulas.
Applications
How much fence should you buy to completely enclose your garden?
Apply the formula to calculate the perimeter of a rectangle: P = 2(l + w)
Do you place money in account A for 6 months with 8% interest earned, or in
account B for 12 months with 7% interest earned? Which will earn more money?
Apply the formula to calculate simple interest: I = Prt
Based on the length of the skid marks, was the car speeding when the brakes were
applied?
Apply the skid to stop formula: S = 30 Df
Chapter 2 — Solving Equations
96
There are many other types of algebraic equations: linear, absolute value, fractional, radical, exponential,
and quadratic, to name a few. Each type of equation has one or more special techniques used to solve for
the unknown, but all equations use the same properties and principles listed below.
To solve an algebraic equation: if all but one variable is known, substitute (replace) the given values
for their matching variables and use the Order of Operations to simplify each side. Solve for the unknown
by applying the appropriate properties of equality.
Properties of Equality
Addition Property of Equality
Multiplication Property of Equality
Algebraic Statement
Example
Algebraic Statement
Example
if a = b then
if x = 3 then
if x = 3, then
a+c=b+c
x+2=3+2
or x + 2 = 5
if a = b then
a $c = b$c
KEY OBSERVATIONS
The rule is add the same quantity (positive or negative
value) to both sides of an equation, but you can also
subtract the same quantity from both sides. Think
about how you change subtraction to addition of the
opposite and apply that process to equations.
x•2=3•2
or 2x = 6
For division (multiplication by the reciprocal)
if a = b then
a b
=
c c
if 5x = 15, then
5 x 15
=
5
5
5 x 15
=
5
5
x=3
Transitive Property of Equality
3
Algebraic Statement
Example
KEY OBSERVATIONS
if a = b and b = c
then a = c
if a = b and b = 7
then a = 7
Multiplying each side of an equation by the same
quantity (except 0) results in an equivalent equation.
Dividing is also accepted under this property, because
division is multiplication by the reciprocal.
KEY OBSERVATIONS
This property is used for substituting one value for
another and in formal proofs.
Reflexive Property of Equality
Algebraic Statement
Example
a=a
1 = 1, 4 = 4
KEY OBSERVATIONS
Any quantity is equal to itself. This property is most
often used in formal proofs.
Symmetric Property of Equality
Algebraic Statement
Example
if a = b then
x = 3 and 3 = x
are equivalent
equations
b=a
KEY OBSERVATIONS
The Symmetric Property of Equality allows interchange
of the left side and right side of an equation. Most western
cultures read from left to right, and so, by convention,
we like the variable to be on the left side of the equal
sign, but it is also correct to have it on the right side.
Section 2.1 — Equations
97
Models
Model 1: The Properties of Equality
Solve the following equations for x. Note which Properties of Equality were used. Validate by substituting
your answer for the variable (x) in the original equation.
Equation
Solve for x
Property or Properties
Used
8 + x = 23
8 + x = 23
8 + (–8) + x = 23 + (–8)
x = 15
Addition Property
(as subtraction —
addition of the opposite)
?
8 + (15) = 23
92 = x – 14
92 + 14 = x – 14 + 14
106 = x
x = 106
Addition
?
92 = (106) – 14
92 = x – 14
3x = 29.7
x
= 11
13
76 = 5x
and
Symmetric
Properties
Validate
23 = 23 
92 = 92 
3 x = 29.7
3 x 29.7
=
3
3
3 x 29.7
=
3
3
x = 9.9
Multiplication Property
(as division —
multiplication by the
reciprocal)
29.7 = 29.7 
x
= 11
13
x
13 : = 11 : 13
13
13 x
= 11 : 13
13
x = 143
Multiplication
Property
143 ?
= 11
13
76 = 5 x
5 x = 76
5 x 76
=
5
5
5 x 76
=
5
5
x = 15.2
Symmetric
?
3(9.9) = 29.7
11 = 11 
and
Multiplication Property
(as division —
multiplication by the
reciprocal)
?
76 = 5(15.2)
76 = 76 
Chapter 2 — Solving Equations
98
Model 2
Solve the following equations for the indicated variable. Use what you have learned about solving algebraic
equations and the Properties of Equality. Validate by substituting your answer and the given value(s) back
into the original equation.
Equation
4x + 7y = –8
Solve
Given
for
value(s)
y
x = –3
Solution
4 x + 8 y = -8
4(-3) + 8 y = -8
-12 + 8 y = -8
-12 + 12 + 8 y = -8 + 12
8y = 4
Validation
4 x + 8 y = -8
?
4(-3) + 8(0.5) = -8
?
-12 + 4 = -8
-8 = -8

8y 4
=
8
8
4
y = = 0.5
8
5x + y = 2z –1
n2 +
r
= 72
2
y
r
x=3
z = –2
n=6
5 x + y = 2 z -1
5(3) + y = 2(-2) -1
15 + y = -5
15 − 15 + y = -5 - 15
y = -20
r
= 72
2
r
(6) 2 + = 72
2
r
36 + = 72
2
r
36 - 36 + = 72 - 36
2
r
= 36
2
r
2
= 2 : 36
2
r = 72
n2 +
5 x + y = 2 z -1
?
5(3) + (-20) = 2(-2) -1
?
15 - 20 = -5
-5 = -5

r
= 72
2
(72) ?
( 6) 2 +
= 72
2
n2 +
?
36 + 36 = 72
72 = 72

Section 2.1 — Equations
99
Methodologies
Solving Proportional Equations
►
Example 1:
5.6 17
=
Round as needed.
x 31
►
Example 2:
17 75.1
=
Round as needed.
x 58.3
Steps in the Methodology
Step 1
Locate the
position of
the unknown
Step 2
Cross multiply
so that the
variable or
unknown is
on the left
Be aware of where the variable
is located to help eliminate
possible errors in alignment
and cross multiplying in Step 2.
Cross multiplying is a short cut
for solving proportions. It uses
the multiplication property
twice in the same step,
resulting in the elimination of
denominators in the ratios.
Use the properties so that the
resulting equation is in the
form you need for solving the
equation.
Try It!
Example 1
Example 2
5.6 17
=
x
31
5.6
17
=
x
31
17 x = 173.6
Note: 173.6 = x17 is correct,
but is more commonly
written as 17x = 173.6 by
applying the commutative
property of multiplication
and the symmetric property
of equality.
Validate the multiplication
with your calculator:
173.6
= 5.6
31
Step 3
Divide by the
coefficient of
the variable
Use the Multiplication Property
of Equality to divide both
sides by the coefficient of the
variable or multiply by the
reciprocal of the coefficient.
17 x 173.6
=
17
17
Answer:
x = 10.21 rounded to two
places.
Validate the division
with your calculator:
10.21 × 17 = 173.57
. 173.6
Step 4
Validate
There are many ways to
validate the solution. One
method is to treat each side
of the equation separately;
perform each division, then
compare results.
Another is to check the equality
of the cross-products.
Divide each side:
5.6 ? 17
=
10.21 31
0.548 ≈ 0.548

0.548 �
c 0.548
Chapter 2 — Solving Equations
100
Models
Model 3
Solve the four proportion equations given in the Proportions section of Building Mathematical Language.
Round to two decimal places.
Solve for x
x 5 x 55x 385x5 538 53 385x 3 38 x13.5 x3.3
13.5 x 31.33.5 13..3
53 3.33
=
= ==
=
== = =
== = =
=
=
24 7 24 7x24 70724x 7705x.7 7021
x5.770 21
5.78 521
.7x 8 21 x8
8x
x
Step 1
Locate the
variable
Step 2
Cross multiply
7x = 120
38x = 350
5.7x = 63
1.5x = 26.4
Step 3
Divide by the
coefficient of x
7 x 120
=
7
7
38 x 350
=
38
38
5.7 x 63
=
5.7
5.7
1.5 x 26.4
=
1.5
1.5
x . 17.14
x . 9.21
x . 11.05
x = 17.6
Step 4
Validate
x 5 x 55x 385x5 538 53 385x 3 38 x13.5 x3.3
13.5 x 31.33.5 13..3
53 3.33
=
= ==
=
== = =
== = =
=
=
24 7 24 7x24 70724x 7705x.7 7021
x5.770 21
5.78 521
.7x 8 21 x8
8x
x
17.14 ? 5
=
24
7
5 ? 38
=
9.21 70
3 ? 11.05
=
5.7
21
1.5 ? 3.3
=
8 17.6
0.7142 . 0.7143 0.5429 . 0.5429 0.5263 . 0.5262  0.1875 = 0.1875 




Model 4
Convert the following measurements using proportion equations. Round to one decimal place.
Conversion ratio:
6 inches to feet
52 inches to feet
0.8 feet to inches
1 foot
12 inches
12 inches
1 foot
12 inches
1 foot
The key to setting up a proportion is to make
sure that the ratios are symmetrical. This means:
Step 1
Locate the
variable
Step 2
Step 3
Step 4
feet
feet
inches inches
=
or
=
inches inches
feet
feet
x feet
1 foot
=
6 inches 12 inches
52 inches 12 inches
=
x feet
1 foot
x inches 12 inches
=
0.8 feet
1 foot
Cross multiply
12x = 6
12x = 52
x = 9.6
Divide by the
coefficient of x
12 x 6
=
12 12
x = 0.5 feet
12 x 52
=
12 12
x . 4.3 feet
0.5 ? 1
=
6 12
0.0833 . 0.0833 
52 ? 12
=
4.3 1
12.09 . 12 
Validate
no coefficient
x = 9.6 inches
9.6 ? 12
=
0.8 1
12 = 12 
Section 2.1 — Equations
101
Model 5
15
Similar triangles: Given that the two triangles are similar,
find the missing side measures. Round to the nearest tenth.
7
25
b
Find Side b
d
30
Find Side d
Again, the key to setting up a proportion is identifying symmetical ratios. If two triangles
are similar, their corresponding angles are equal, so we can match shortest sides, longest
sides, and middle-sized sides.
Set up the ratios so that you can tell which side belongs to which triangle:
smaller triangle side
larger triangle side
Step 1
Locate the variable
Step 2
Cross multiply
Step 3
Divide by the
coefficient of x
Step 4
Validate
15 corresponds with 25
7 corresponds with d
b corresponds with 30
15 corresponds with 25
7 corresponds with d
b corresponds with 30
15 b
=
25 30
15 7
=
25 d
25b = 15 × 30
25b15
= 15
× 30450
: 30
b=
=
15 25
: 30 450
25
b=
=
25
b = 18 25
b = 18
15d = 25 × 7
15d 25
= 25: 7× 7175
d=
=
2515: 7 175
15
d=
=
15
15
d . 11.7
d . 11.7
15 ? 18
=
25 30
?
15 × 30 =
25 × 18
15 ? 7
=
25 11.7
175.5 . 175 
450 = 450 
Chapter 2 — Solving Equations
102
Methodologies
Setting up and Solving Percent Equations
►
Example 1:
12.5% of what number is 420?
►
Example 2:
13% of what number is 53?
Steps in the Methodology
Step 1
Write the
unknown
number as a
variable
Percent equations take the
form of: percent of a base is an
amount.
Try It!
Example 1
12.5% of B is 420
The base is the original or whole
quantity; the amount is a part of
or change from the original base.
We will use variables as follows:
B = base
P = percent
A = amount
The unknown will usually be
presented with the word what
as in “what number” or “what
percent” or “is what amount.”
The word “find” also translates to
“what is.” Look for word patterns
to guide you.
Step 2
Change “of”
to multiply
and “is” to
equal
The word of always indicates
multiplication in the context of
percent equations.
12.5% • B = 420
For example:
50% of 12 means 50% • 12
The word is always translates
to “=”. Once the words are
translated, an equation is ready
to be solved.
Step 3
Change
percent
notation
to decimal
notation.
Review changing from percent to
decimal: move the decimal point
two places to the left (divide by
100) and drop the % sign.
Special
Case:
For Type 2 percent
problems, skip this step
and proceed to Step 4.
0.125 • B = 420
Example 2
Section 2.1 — Equations
103
3.Example
2 2.8 1
=
12.8
x
Steps in the Methodology
Step 4
Solve
for the
variable by
multiplying
or dividing.
Step 5
Validate
Solve the equation for the
unknown.
Example 2
0.125 B
420
=
0.125 0.125
For Type 2 percent
problems: to present the
answer as a percent,
Special
move the decimal point
Case:
two places to the right and
attach a percent sign (%).
(See Model 6, Type 2.)
Use the Percent × Base = Amount
equation. Substitute your answer
and one of the given values in
the appropriate places. Solve the
resulting equation to match the
other original given value.
Answer:
x = 3360
B = 3360
Percent × Base = Amount
Found base (3360), use
amount (420) and solve for
percent:
P • 3360 = 420
P
3360
3360
=
420
3360
P = 0.125
change to percent:
12.5% 
Note: Sometimes Steps 1 and 2 are not necessary. If the problem is presented in an equation form, proceed
with Steps 3 through 5.
Model 6
Type 1
Percent Equations: Translate into an equation and multiply or divide to solve.
What is 15% of 27.50?
Step 1
Change to A
A is 15% of 27.50
Step 2
Translate
A = 15% • 27.50
Step 3
Convert percent to decimal
A = 0.15 • 27.5
Step 4
Solve
A = 4.125
Step 5
Validate
Percent × Base = Amount
Found amount (4.125), use percent (0.15) and solve for
base: 0.15 • B = 4.125
0.15 4.125
=
0.15
0.15
B = 27.5 
B
Chapter 2 — Solving Equations
104
Type 2
7.85 is what percent of 35.7?
Step 1
Change to P
7.85 is P of 35.7 (Note: P will be a percent in its decimal form.)
Step 2
Translate
7.85 = 35.7P
Step 3
Convert percent to
decimal
OMIT
Step 4
Solve
7.85
=P
35.7
P ≈ 0.22
change to %: P ≈ 22%
Step 5
Validate
Percent × Base = Amount
Found percent (0.22), use amount (7.85) and solve for base:
0.22 • B = 7.85
0.22 7.85
=
0.22 0.22
B . 35.68 . 35.7 
B
Type 3
Find B if 93 = 18% • B
Step 3
Convert percent to decimal
93 = 0.18B
Step 4
Solve
93
=B
0.18
516.67 ≈ B
Step 5
Validate
Percent × Base = Amount
Found base (516.67), use percent (0.18) and solve for
amount: 0.18 • 516.67 = A
A ≈ 93 
Section 2.1 — Equations
105
Addressing Common Errors
Issue
Not changing
a percent to
a decimal
before solving
a percent
equation
Incorrect
Process
Resolution
12% of what number Whenever a number
is presented in
is 3?
12B = 3
B = 1/4
percent form, make
sure to rewrite it as
a decimal (Step 3
in the Methodology
for Solving Percent
Equations).
Correct
Process
0.12 B = 3
3
B=
0.12
B = 25
What percent of 123
is 42?
P : 123 = 42
123P = 42
42
P=
123
P = 0.34
Always check the
calculated solution
of the equation with
the required form
of answer for the
problem.
This is the special
case within percent
equations.
P : 123 = 42
123P = 42
42
P=
123
P . 0.34
so P . 34%
3x – 3 = 21
3x +3 – 3 = 21
3x = 21
x=7
Think about an
equation as a
balance scale: keep
the sides balanced
by adjusting the
scale on both sides
an equivalent
amount.
Found base (25),
use amount (3) and
solve for percent:
(P) • 25 = 3

25
3
=
25 25
P = 0.12
change to percent:
P = 12% 
This is the decimal
form of the percent.
Multiplying
or adding
a quantity
to only one
side of an
equation
P×B=A
P
Note that this
only applies when
solving as a percent
equation, not when
solving a percent
equation as a
proportion.
Not changing
the answer to
percent when
required
Validation
3 x - 3 = 21
3 x + 3 - 3 = 21 + 3
3 x = 24
3 x 24
=
3
3
x =8
P×B=A
Found percent
(0.34), use base
(123) andsolve for
amount:
0.34 • 123 = A
A = 41.82 ≈ 42 


?
3(8) – 3 = 21
?
24 – 3 = 21
21 = 21 

Chapter 2 — Solving Equations
106
Issue
Not validating
the answer in
the original
equation
Incorrect
Process
Validate the solution Validation should be
performed with the
x = 7 for the
original equation
following work
3x – 3 = 21
3x +3 – 3 = 21
3x = 21
x=7
only.
Answer: Substitute
7 for x in 3x = 21:
3(7) = 21 
Not
identifying
the correct
amount in
a percent
problem
12 is what percent of
48?
P : 12 = 48
48
P=
=4
12
P = 400%
7 does not check when the original
equation is used for validation.
Substitute x = 7 into 3x – 3 = 21
NOT into the equation from Step 3
(3x = 21).
3x – 3 = 21
?
3(7) – 3 = 21
?
21 – 3 = 21
18 ≠ 21
Does not validate; go back and find the error.
Translate the
problem exactly as
written replacing is
with = and of with ×
or • (a multiplication
symbol).
The base is
connected to
the percent by
multiplication and
usually follows the
word of.
The amount should
be alone on one side
of the equal sign.
Not
identifying
the correct
base in a
percent
problem
Correct
Process
Resolution
30% of what number Same resolution as
previous issue.
is 15?
12 is what percent of 48?
12 = P • 48.
or 12 = P × 48
12 = 48 P
12
=P
48
0.25 = P
25% = P
P = 25%
P×B=A
30% of what number is
15?
P×B=A
.30 × 15 = x
4.5 = x
0.30B = 15
B = 50
Found percent
(0.25), use base
(48) and solve for
amount:
(0.25) • 48 = A
A = 12 
Found base (50),
use percent (0.30)
and solve for
amount:
(0.30) • 50 = A
A = 15 
Preparation Inventory
Before proceeding, you should be able to do each of the following:
Apply principles and properties of equations.
Correctly interpret information to create symmetrical ratios in a proportional equation.
Solve proportion equations.
Solve percent equations.
Find the value of an unknown variable in an algebraic equation.
Section 2.1
Activity
Equations
Performance Criteria
• Setting up and solving proportion equations
– correct identification of corresponding parts
or units
– formation of correct proportions
– correct application of the methodology for
solving proportions
– accuracy of calculations to the designated
decimal place
– validation of the answer
• Solving an algebraic equation containing more
than one variable, when given values for the
other variables
– correct substitution of values
– correct application of the Order of
Operations
– accuracy of calculations
– validation of the answer
• Setting up and solving percent equations
– correct identification of amount, base, and
percent
– correct application of percent notation
– accuracy of calculations
– validation of the answer
Critical Thinking Questions
1. What are four ways to change an equation without changing the value(s) of the variable(s) that make it
true?
2. Why is it important to always add the same quantity to both sides of an equation when solving an
equation?
107
Chapter 2 — Solving Equations
108
3. Why do you need to change from percent notation to decimal notation before calculations?
4. How can you tell the difference between the base and the amount in a percent problem so you can write a
correct percent equation?
5. How do you validate a solution to an equation?
Tips
for
Success
• Validate arithmetic processes each step of the way
• The preferred form for the answer is to rearrange the final equation so that it reads from left to right:
x = answer
• Validate not only the answer, but also to see if the form of the answer is correct—is the number of decimal
places correct? Or does the answer need to be converted to a percent?
• Before solving similar triangles, be sure that you have lined them up correctly, even redrawing them if
necessary, in order to identify corresponding sides and angles
Section 2.1 — Equations
109
Demonstrate Your Understanding
1. Solve the following equations for x. Note which Properties of Equality were used. Validate by substituting
your answer for the variable (x) in the original equation.
Equation
a)
x – 19 = 21
b)
240 = x + 189
c)
2x +1 = 33
d)
814 =
Solve for x
Property or
Properties Used
Validate
x
6
2. Solve each algebraic equation for the indicated variable:
Problem
a)
3x + y = 6
if x = –2
for y
Worked Solution
Validation
Chapter 2 — Solving Equations
110
Problem
b)
x + 7y =12
Worked Solution
Validation
for x
if y = 1
c)
x
=z
y
for x
if y = 3 and z = –2
d)
3x - y = n + 5
for y
if x = 3 and n = –2
3. Solve the following proportion problems. Round to two places if necessary.
Problem
a)
3 x
=
5 20
b)
3.2 11
=
x 17.5
c)
x
84.7
=
5.6 168.3
Answer
Validation
Section 2.1 — Equations
111
4. Using the following conversion ratios, solve for the indicated measurement. Round to the nearest tenth.
Conversion
Ratio
Problem
a)
62 inches is
how many
feet?
1 foot
12 inches
b)
81 feet is
how many
inches?
1 foot
12 inches
c)
7.5 yards is
how many
feet?
1 yard
3 feet
d)
16 feet is
how many
yards?
1 yard
3 feet
Answer
5. Use the diagram below to solve the following problems. Triangle
ABC is similar to triangle DEF. Round to the nearest tenth.
Validation
a
b
c
Problem
a)
Find side e if a = 6,
f = 1.5, and b = 9.
b)
If f = 12, d = 18 and
c = 20, find side a.
Worked Solution
f
e
d
Validation
Chapter 2 — Solving Equations
112
6. Solve the following percent equations:
Problem
a)
12% of what number
is 42?
b)
15 is what percent of
85?
c)
16.5% of 86 is what
amount?
d)
What percent of 85
is 98.6?
e)
12.5% of 52 is what
amount?
Equation
Answer
Validation
Section 2.1 — Equations
Identify
and
113
Correct
the
Errors
In the second column, identify the error(s) in the worked solution or validate its answer. If the worked solution
is incorrect, solve the problem correctly in the third column and validate your answer.
Worked Solution
1)
Solve for x:
12 3
=
x 11
3x = 132
x = 44
2)
12% of 45 is what
amount?
12 × 45 = 540
3)
What percent of 80 is
120?
x × 120 = 80
80
x=
120
x = 67%
Identify Errors
or Validate
Correct Process
Validation
Chapter 2 — Solving Equations
114
Worked Solution
4)
Solve for z in the
equation below if y = 2,
and x = 5.
9x + 3y =
z
4
Answer:
9(5) + 3(2) =
45 + 6 =
z
4
z
4
z
4
51 4 z
=
4
4
51
z = = 12.75
4
51 =
5)
Solve for x when y = 2
x – y2 = 17
x – 22 = 17
x + 4 = 17
x + 4 – 4 = 17 – 4
x = 13
Identify Errors
or Validate
Correct Process
Validation