Linear Equations with
One Variable
Connections
Have you ever . . .
• Calculated your wages including commission or tips?
• Decided how much to charge for a product based on cost
of production?
• Determined the cost to take a cab to your destination?
Although you may not be aware of them, linear equations are
common in our day-to-day lives and in the workplace. How would
you write an equation for wages plus commission? What about an
equation for the taxi rates in your town?
A linear equation can be graphed as a straight line. A linear equation:
• Never has a variable in the denominator of a fraction or divides by the variable.
• Never has a variable with an exponent (such as x2 or q4).
A linear equation with one variable has only one type of variable (such as x or q), even if it
appears more than once. It creates a straight horizontal or vertical line on a graph with two
axes because the value of the variable is always the same. Here are three examples:
1
7
7x - 1 = 13 -y - 2 = -6 3 x - 4 = 4 + 2x
x
x
x
y
y
y
To solve a linear equation with one variable, isolate the variable. Move all the constants to
one side of equation, with the variable on the other side by itself.
117
Essential Math Skills
Learn
It!
Solving Linear Equations with Undo
Use the undo strategy to isolate the variable. To undo an algebraic equation:
• Undo parentheses by simplifying and distributing.
• Multiply and divide constants and terms.
• Combine like terms.
• Use inverse operations to undo addition and subtraction.
• Use inverse operations to undo multiplication and division.
When you’re done, the variable will be isolated on one side of the equation, and a number
will be on the other. That’s your solution.
Kata earns $250 for every website she designs plus a bonus of 3% of $55 for each
referral from her websites. Last month, she also earned a $130 month-end bonus.
Kata had three referrals from each of her websites and earned $3,699.30. How
many websites did she design?
Using U n PAC , you can write a linear equation with one variable to solve the
problem. Let w = the number of websites:
250w + 0.03(55 ◊ 3w) + 130 = 3699.3
Undo Parentheses
To undo the parentheses, simplify the amount in the parentheses if
possible. Then, distribute any amount multiplied by the parentheses.
The distributive property states that a (b + c) is equal to ab + ac. If you
pay 8.5% tax on a $12 shirt and $23 pants, you’re paying the same tax
whether you buy them together or separately.
0.085(12 + 23) = 0.085(12) + 0.085(23)
The distributive property is true because multiplication is actually
repeated addition:
3(2 + 4) = (2 + 4) + (2 + 4) + (2 + 4) = (2 + 2 + 2) + (4 + 4 + 4) = (3)(2) + (3)(4)
Using
Un P A C
Undo is a strategy
to attack the
problem by solving
an algebraic
equation.
derstand
lan
ttack
heck
118
Linear Equations with One Variable
Undo the parentheses in the example problem.
?1.
Since 3w means 3 ◊ w, you can simplify the parentheses by multiplying 55 times three.
250w + 0.03(55 ◊ 3w) + 130 = 3699.3
250w + 0.03(165w) + 130 = 3699.3
Now multiply 0.03 times 165 to distribute it and remove the parentheses.
250w + 4.95w + 130 = 3699.3
Multiply or Divide Constants and Terms
A term is a constant with or without variables, such as 5x, 420, or 2xy. The constant in a
term with a variable is called a coefficient. To multiply a term with a variable by a constant,
multiply the coefficient by the constant.
2x ◊ 12 = 2 ◊ x ◊ 12 = 24 ◊ x = 24x
You’ve already multiplied a constant and a term by multiplying 55 by 3w.
You can also divide a term by a constant. Divide the coefficient by the constant.
2
1
2x ' 12 = 2 # x ' 12 = (2 ' 12) x = 12 x = 6 x
There are no more terms to multiply or divide in the example equation.
Combine Like Terms
To combine like terms, add and subtract terms with the same variables by adding or
subtracting the coefficients.
Combine like terms.
?2.
The like terms are 250w and 4.95w. You can combine them by adding the coefficients.
250w + 4.95w + 130 = 3699.3
254.95w + 130 = 3699.3
119
Essential Math Skills
Undo Addition and Subtraction
Undo addition and subtraction using the inverse operation.
Undo addition and subtraction in the example equation.
?3.
There is no subtraction in the example equation, but there is a number added to the
variable: 130. Subtract it from each side of the equation.
254.95w + 130 = 3699.3 -130 -130 254.95w = 3569.3
Undo Multiplication and Division
Undo multiplication and division using the inverse operation. You should end up with the
variable isolated on one side of the equation.
Undo multiplication to isolate the variable in the example equation.
?3.
Divide by 254.95 to undo multiplication and isolate the variable w.
254.95w = 3569.3
254.95w 3569.3
254.95w ÷ 254.95 = 3569.3 ÷ 254.95 also written as 254.95 = 254.95
w = 14
Kata designed 14 websites. You can graph w = 14 as a line where w always equals 14.
w
16
14
12
10
8
6
4
2
0
120
x
Linear Equations
Lesson
with
Title
One Variable
e
ic
Pract
It!
Use your knowledge of linear equations with one variable to complete the
following exercises.
1.Circle the linear equations.
x
1
5x - 4 x + 5 = 10 - x 7d = 14 a2 + 7 = 16 2 = 10
Explain how to determine if an equation with one variable is linear.
2.-2 - 3b + 5 = 17 + b
a.
Use undo to solve the equation. b.
A linear equation
Show and explain each step. graphs as a line on
a graph with two
axes. Graph the
solution to this
equation to show
the line.
b
5
4
3
2
1
–5 –4 –3 –2 –1
–1
–2
–3
–4
–5
0
x
1 2 3 4 5
3.Blanca made 24 black and white photocopies. She then spent $4.00 on color copies. Her
total came to $4.72.
Write and solve a linear equation to find how much each black and white copy costs.
Show and explain each step.
4.Albert bought four bamboo plants for his company and spent $24 on a Japanese maple
tree for himself. He spent a total of $82. He needs to know the cost of the bamboo
plants so that he can get reimbursed.
Write and solve a linear equation to find the price of each bamboo plant. Show and
explain each step.
121
Essential Math Skills
5.An economist identifies the following equation to find the price point where supply
equals demand for a product. In economics, the variable P is often used for price.
-88 + 2P = 152 - 2P
Simplify and solve this equation for P, explaining each step. Remember that you can
add and subtract terms by adding and subtracting the coefficients.
6.Use your understanding of algebra, multiplication, and addition to explain why 3x plus
2x equals 5x.
7.Simplify the following expressions and solve the following equations.
a.
3x + 1.5 (13 + x) + xb.
2x + 9x - x + 0.5x - 6.5 - 3x + 12
c. 6(4.2 + x - 3) + 5(x + 2x + x - 3)d.
x + 6(12 )( 2x) = 32
e.
2x + 9x - x = 36 - 2xf.
4(2 + x) + 2(3x + x) = 80
8.Demonstrate and explain the distributive property using an original example equation.
Math Tip
9.Why is simplifying expressions important in
solving algebra problems?
A negative sign must stay with its
number. To keep track of negatives,
use parentheses.
You can write:
2(3x - 4x + 10x - 2x + x)
as:
2[(3x) + (-4x) + (10x) + (-2x) + (x)]
122
Linear Equations with One Variable
Check Your Skills
Using your knowledge of linear equations, answer the following questions.
1.
Mateo works in a sporting goods store. He earns $10.15 an hour plus an 8%
commission on every dollar he sells. Last month he worked 120 hours and made
$1,410 total. How much were his total sales in dollars last month?
a.
$1,354
b.
$1,600
c.
$1,763
d.
$2,400
y
2.Which of the following equations does
this graph represent?
7
6
5
4
3
2
1
a.
2y = 3
b.
y = 2.5
c.
2y = -5
d.
-2y = -3
–7 –6 –5 –4 –3 –2 –1
–1
–2
–3
–4
–5
–6
–7
0
x
1 2 3 4 5 6 7
3.
Kim bought a computer and a keyboard. She paid a total of $1,554. She received a
student discount of 20% on the computer but not the keyboard, which cost $70.
What was the original price of the computer?
a.
$1,498
b.
$1,855
c.
$1,925
d.
$2,100
4.Solve for y.
6 + 2(y + 8) - 3.5y = 12 + y
123
Essential Math Skills
5.Yvette plans to build a shop cabinet. She will need 64 feet of two-by-fours at $3.50 a
foot plus hardware. If Yvette’s budget is $300, how much can she spend on hardware?
a.
$64
b.
$72
c.
$76
d.
$80
6.
Victor owns a small knitting business. The price of yarn is 3.5 cents ($0.035) per
yard, and a pom pom costs 10 cents. He has a special order for pom pom hats
with scarves. The hats require 150 yards of yarn and five pom poms, and the scarves
require 250 yards of yarn. If Victor charges $125, what is Victor’s profit?
7.Annette has started a business selling stools at a local craft market. Each stool costs
$2.50 to make, and she pays $50 a day to rent a vending booth. Each stool sells for $25.
Last Saturday, Annette made $287.50 in profit. How many stools did she sell?
a.
10
b.
12
c.
15
d.
25
8.Which of the following is a simplification of this equation?
2x + 3(32x + 7) = 12
a.
96x + 7 = 12
Remember
the Concept
b.
96x - 7 = 12
• Undo parentheses.
c.
98x - 21 = 12
• Multiply and divide
constants with
terms.
d.
98x + 21 = 12
• Combine like terms.
• Undo addition and
subtraction.
• Undo multiplication
and division.
124
Answers and Explanations
Linear Equations with
One Variable
page 117
Solving Linear Equations with Undo
Practice It!
3.
Let c = the price for a black and white copy. Blanca
bought 24 black and white copies (24c) then paid an
additional $4 for the color copies. Her total was $4.72.
24c + 4 = 4.72
pages 121–122
1
- x + 5 = 10 - x
1.
4
There are no parentheses or terms to multiply or
divide. There are also no like terms to combine. So
first, use inverse operations to move the added four.
7d = 14
24c + 4 - 4 = 4.72 - 4
x
= 10
2
24c = 0.72
A linear equation with one variable must be an equation, not an expression. It cannot have any variables
with exponents or numbers divided by a variable. A
linear equation will graph a straight vertical or horizontal line.
Next, use inverse operations to move the 24.
2a.b = -3.5
4.
Each bamboo plant was $14.50.
There are no parentheses or multiplication or division
of terms. So first, combine like terms:
-2 - 3b + 5 = 17 + b
24c ÷ 24 = 0.72 ÷ 24
c = 0.03
Black and white copies are three cents a page.
Let b = the cost of one bamboo plant. Albert bought
four bamboo plants (4b). He also spent $24 on
another kind of plant. His total was $82.
-3b + 3 = 17 + b
Next, use inverse operations to get the variables on
one side and the constants on the other.
Use inverse operations to move the 24. Subtract
24 from each side.
-3b + 3b + 3 = 17 + b + 3b
4b + 24 - 24 = 82 - 24
3 = 17 + 4b
4b = 58
3 - 17 = 17 - 17 + 4b
Use inverse operations to move the four. Divide each
side by four.
-14 = 4b
Finally, use the inverse operations of multiplication
and division to isolate the variable.
-14 ÷ 4 = 4b ÷ 4
4b + 24 = 82
4b ÷ 4 = 58 ÷ 4
b = 14.5
5.
P = 60
-3.5 = b
-88 + 2P = 152 - 2P
2b.
There are no parentheses or terms to multiply and
divide. There are also no like terms on the same side
of the equation. Use inverse operations to move all
the variables to one side of the equation.
b
5
4
3
2
1
–5 –4 –3 –2 –1
–1
–2
–3
–4
–5
-88 + 2P + 2P = 152 - 2P + 2P
x
0
1 2 3 4 5
-88 + 4P = 152
Use inverse operations to move all added
and subtracted constants to the other side of
the equation.
-88 + 88 + 4P = 152 + 88
4P = 240
i
Essential Math Skills
Use inverse operations to move the coefficient to the
other side of the equation and isolate the variable.
4P ÷ 4 = 240 ÷ 4
P = 60
6.
The term 3x is the same as three times x.
3x + 2x = 3:x + 2:x
Since multiplying a times b is the same as adding a to
itself b times, you can rewrite the expression as the
variable added to itself five times:
3:x + 2:x = (x + x + x) + (x + x)
7.2 + 6x + 20x - 15
Combine like terms:
26x - 7.8
7d.To solve this equation:
x + 6(12)(2x) = 32
Simplify the parentheses by multiplying terms:
x + 6(24x) = 32
Distribute the number multiplied by the parentheses:
The variable added to itself five times is the same as
five times the variable.
x + 144x = 32
x + x + x + x + x = 5:x = 5x
145x = 32
7a.To simplify this expression:
Divide by 145 to isolate the variable:
3x + 1.5(13 + x) + x
145x ÷ 145 = 32 ÷ 145
First, distribute 1.5 to remove the parentheses:
x = 0.22068 . . .
7e.To solve this equation:
3x + 1.5(13) + 1.5x + x
Combine like terms:
3x + 19.5 + 1.5x + x
2x + 9x - x = 36 - 2x
Next, combine like terms:
Combine like terms:
10x = 36 - 2x
3x + 1.5x + x + 19.5
4.5x + x + 19.5
5.5x + 19.5
Use inverse operations to move the 2x to the left side
of the equation:
7b.To simplify this expression:
10x + 2x = 36 - 2x + 2x
2x + 9x - x + 0.5x - 6.5 - 3x + 12
12x = 36
Change subtraction to plus a negative to make
combining like terms easier:
Use inverse operations to move the 12 to the right
side of the equation:
2x + 9x + (-x) + 0.5x + (-6.5) + (-3x) + 12
Move like terms together to make combining like
terms easier:
x = 3
2x + 9x + (-x) + 0.5x + (-3x) + 12 + (-6.5)
4(2 + x) + 2(3x + x) = 80
Combine like terms:
Simplify the parentheses by combining like terms:
7.5x + 5.5
4(2 + x) + 2(4x) = 80
7c.To simplify this expression:
Distribute the numbers multiplied by the parentheses:
6(4.2 + x - 3) + 5(x + 2x + x - 3)
8 + 4x + 8x = 80
Simplify the expressions in parentheses by combining
like terms:
Combine like terms:
6(1.2 + x) + 5(4x - 3)
ii
Distribute the numbers multiplied by the parentheses:
12x ÷ 12 = 36 ÷ 12
7f.To solve this equation:
8 + 12x = 80
Answers and Explanations
Use inverse operations to move the eight to the right
side of the equation:
8 - 8 + 12x = 80 - 8
Use inverse operations to move the 12 to the right
side of the equation:
10 = 2.5y
5.
c. $76
x = 6
8.
The distributive property says that a(b + c) = ab + ac. If
you bought two packs of three toys and two packs of
six toys, you’d have the same amount of toys no matter whether you bought them together or separately:
2(3 + 6) = 2(3) + 2(6)
9.
Simplifying algebraic expressions is important in solving algebra problems because it allows you to deal
with fewer constants and terms. It is easier to isolate
the variable in an equation where the expressions on
each side have been simplified.
pages 123–124
1.
d. $2,400
Let s = total sales.
10.15(120) + 0.08s = 1410
Multiply terms:
1218 + 0.08s = 1410
Use inverse operations to move the 1218:
1218 - 1218 + 0.08s = 1410 - 1218
0.08s = 192
Use inverse operations to move the 0.08:
s = 2400
22 = 12 + 2.5y
4 = y
12x ÷ 12 = 72 ÷ 12
0.08s ÷ 0.08 = 192 ÷ 0.08
6 + 2y + 16 - 3.5y = 12 + y
-1.5y + 22 = 12 + y
12x = 72
Check Your Skills
6 + 2(y + 8) - 3.5y = 12 + y
(64)(3.5) + x = 300
224 + x = 300
x = 300 - 224 = 76
6.
$110.50
(150)(0.035) + (5)(0.1) + (250)(0.035) = 125 - x
5.25 + 0.5 + 8.75 = 125 - x
5.75 + 8.75 = 125 - x
14.5 = 125 - x
14.5 + x = 125
x = 125 - 14.5 = 110.5
7.
c. 15
Let s = the number of stools sold.
s(25 - 2.5) - 50 = 287.5
22.5s - 50 = 287.5
22.5s = 337.5
s = 15
8.
d. 98x + 21 = 12
2x + 3(32x + 7) = 12
2x + 96x + 21 = 12
98x + 21 = 12
2.
c. 2y = -5
3.
b. $1,855
Let p = original price of the computer
p - 0.2p + 70 = 1554
0.8p + 70 = 1554
0.8p = 1484
p = 1855
4.
y=4
iii