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Algebra Notes
Chapter 3 – Solving Linear Equations
Section 3.1 – 3.2 Solving Linear Equations
Goals:
 Identify linear equations
 Solve linear equations using addition and subtraction
 Solve linear equations using multiplication and division
 Identify Properties of Equality
Vocabulary
 Transformations 
Transposition –
Linear Equation
 Variables with a power of 1
 No variables in a denominator
 No variables under a radical sign (√
 No variables in absolute value
)
Steps for Solving Equations
****** Remember to line up the equal signs from one step to the next*******
1. Eliminate double operations
 a + (-b)  a – b
 a – (-b)  a + b
2. Isolate the variable on one side
 Addition  transpose to subtraction
 Subtraction  transpose to addition
 Multiplication  use divison
 Division  use multiplication
 Fractional Coefficients  use multiplication by the reciprocal
3. Simplify
Examples:
a. x – 5 = -13
c. x + 9 = 17
b. -8 = n – (-4)
d. -11 = n + (-2)
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e. -4x = 3
g.
𝑥
5
f. -3x = -5
= -6
i. 10 =
−2
3
h.
m
𝑥
−9
= -5
j. -14 =
−7
8
n
Properties of Equality

Addition Property of Equality

Subtraction Property of Equality

Multiplication Property of Equality

Division Property of Equality
Page 2
Section 3.3 Solve Multi-Step Equations
Goals
 Use two or more transformations to solve an equation
Steps for solve two-step equations
1. Simplify one or both sides of the equation as needed
 Combine like terms
 Distribute
2. Use inverse operations (transformations) to isolate the variable
 Undo addition and subtraction first
 Then undo multiplication and division
Examples:
a. 4x + 3 = 11
b.
1
x – 5 = 10
2
c. -3x + 3 = 18
d.
1
x + 6 = -8
3
e. 7x – 3x – 8 = 24
f. 2x – 9x + 17 = -4
g. 5x - 3(x + 4) = 28
h. 4x + 12(x – 3) = 28
i. 4x – 3(x -2) = 21
j. 2x – 5(x – 9) = 27
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Multiply by a reciprocal first to eliminate fractions
 Sometimes distributing a fraction results in numerous fractions  easier to remove
the fraction first by multiplying each term by the reciprocal
Examples:
a. 66 =
1
2
(x + 3)
b. 12 =
3
10
(x + 2)
c. -24 =
4
3
(x – 7)
Page 4
Section 3.4 Solving Equations with Variables on Both Sides
Goals:
 Collect variables on one side of the equation and then solve the equation
Collect variables on one side
 General rule transpose the smaller coefficient to the larger coefficient’s side
 If only a variable with coefficient is on one side by itself, transpose to that side
Examples
a. 7x + 19 = -2x + 55
b. 6x + 22 = -3x + 31
d. 80 – 9y = 6y
c. 17 – 2x = 14 + 4x
e. 64 – 12 w = 6w
Special Equations
 Variables subtract out leaving constants
o Identity  true statement remains (All real numbers are solutions)
o No Solutions  false statement remains
Examples:
a. 3(x + 2) = 3x + 6
c. 4(x – 5) = 4x + 20
b. x + 2 = x + 4 – 3
d. 3x + 9 = 6x – 3(x -3)
More Complicated Equations
 Remove parenthesis
o Distribute
o Multiply by reciprocal
 Combine like terms
 Collect variables on one side
 Undo addition and subtraction
 Undo multiplication and division
Examples:
a. 4(1-x) + 3x = -2(x + 1)
b.
1
(12x + 16) = 10 -3(x – 2)
4
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c. 10(2 – x) + 4x =
e.
1
2
−3
10
(x + 3)
d.
2
5
(10x + 15) = 18 -4(x – 3)
(12-2x) – 4 = 5x +2(x – 7)
Page 6
Section 3.6 Solving Decimal Equations
Goals:
 Find exact and approximate solutions of equations that contain decimals
 Remove decimals from an equation before solving
Vocabulary
 Round-off error –
Example:
Three people ordered a pizza, which cost $12.89 with tax. They want to share the
cost equally. How much will each pay?
Since we are using money, round to the nearest
cent (hundredth)
Change decimal coefficients to integers before solving
 Multiply EACH term on BOTH sides by a power of 10
Example:
When decimals given in a problem, answer rounds to one place further
a. 4.5 – 7.2x = 3.4x – 49.5
b. 3.7x – 2.5 = 6.1x – 12.2
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Section 3.7 Formulas and Functions
Goals
 Solve a formulas for one of its variables (literal equations)
 Rewrite an equation in function form
 Create a function table for equations after transformation
Vocabulary
 Formula
Examples
a. A = lw
for l
b.
5
(F-32)
9
for F
c. I = prt
for r
Function form
 Solve for y in terms of x (y on one side by itself)
Examples:
Write each of the following in function form (y is a function of x)
a. 3x + y = 4
b. 5x + 2y = 20
Rewrite the above so that x is a function of y (x on a side by itself)
a.
b.
Create a function table for the above functions using x = {-2, -1, 0, 1}
Page 8
Section 3.8 Rates, Ratios, and Percent
Goals:
 Write ratios in proper form
 Calculate unit rates
 Calculate with percents
Vocabulary
 Ratio

Rate

Unit rate

Percent
Ratios in proper form
 Must have 2 numbers
 Must be in the same units
 Must be in simplified form
To find unit rate
 Create a rate with the given information
 Divide and label
Examples:
Below are the total amount spent for several essential items. Find the cost per person
based on a sample size of 266 million people
a. Medical care - $913 billion
b. Housing - $878 billion
c. Transportation - $602 billion
Unit Analysis
 Set up a rate comparing information given
 Multiply with a conversion factor (be sure that unwanted units are at different
levels than the given information rate)
 Unwanted units should divide out leaving only values and wanted units
Examples:
a. Convert $180 to pescos (9.990 pescos = $1)
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b. Convert 180 pesos to dollars (9.990 pesos = $1)
c. $150 to Canadian dollars (1.4 Canadian dollars = $1)
d. 150 Canadian dollars to dollars (1.4 Canadian dollars = $1)
Calculate with percent
part
Percent =
Divide and then change to percent
total
(move decimals 2 places right)
General Formula: Percent of base = percentage (% of total = part)
Examples:
a. Three hundred fifty teenagers were surveyed. About 160 of those surveyed
were dating on a regular basis. What percent of the teens were dating?
b. Average water usage in a household is divided among kitchen (5%), bathroom
(74%), and miscellaneous sources (21%). If the kitchen uses approximately 9
gallons per day, how much it used in an entire house each day?
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