Name: ____________________
Chapter 2 Notes (Solving Linear Functions)
2.1 Writing Equations
Translate the following sentences into algebraic equations.
a)
Five times the number a is equal to three times the sum of b and c.
b)
Nine times y subtracted from 95 equals 37
c)
Eight less than the product of x and 5 is 25.
Translate the following equations into sentences.
a)
3m + 5 = 14
b)
w + v = y2
c)
6 – 2x = 15
Applications
Ex. 1:
The first ice cream plant was established in 1851 by Jacob Fussell. Today, approximately
2,000,000 gallons of ice cream are produced in the United States each day.
In how many days can 40,000,000 gallons of ice cream be produced in the United States?
Ex. 2:
Translate the following sentence into a formula.
The perimeter of a rectangle equals two times the length plus
two times the width.
w
l
Ex. 3:
Mark has $1900 in the bank. He wishes to increase his account to a total of $3500 by
depositing $30 per week from his paycheck. Will he reach his goal in 1 year?
2.2 Solving Equations by Using Addition and Subtraction
Equivalent Equations:
Solve an Equation:
Addition Property of Equality
Words:
If an equation is true and the same number is added to each side of the equal
sign, the resulting equation is also true.
Symbols:
For any numbers a, b, and c, if a = b, then a + c = b + c
Examples:
Subtraction Property of Equality.
Words:
If an equation is true and the same number is subtracted to each side of the
equal sign, the resulting equation is also true.
Symbols:
For any numbers a, b, and c, if a = b, then a - c = b - c
Examples:
Solve the following equations AND check your solutions:
1.
m – 48 = 29
2.
21 + q = -1
3.
142 + d = 97
4.
r – 26 = -2

5.
-12 + f = -8
6.
Write AND solve the equation:
7.
A number increased by 5 is equal to 42. Find the number.
8.
Ninety-eight less than a number is -13. Find the number.
9.
The sum of -16 and a number is 49. Find the number.
m – 123 = 456
2.3 Solving Equations by Using Multiplication and Division
Multiplication Property of Equality:
Words:
If an equation is true and each side is multiplied by the same number, the
resulting equation is also true.
Symbols:
For any numbers a, b, and c, if a = b, then ac = bc
Examples:
Division Property of Equality:
Words:
If an equation is true and each side is divided by the same non-zero number, the
resulting equation is also true.
Symbols:
For any numbers a, b, and c, if a = b, then
Examples:
a b
 (with c not equal to 0)
c c

Solve the following equations. Check your work.
1.

t
7
30
2.

y
 4
6
3.
13d = 143
4.
-41g = -369
5.
 1 
1
2 x  1
 4 
2
6.
22x = -77

Write and solve an equation from each algebraic expression.
7.
Negative eighteen times a number equals -198
8.
The quotient of a number and -8 is 12
2.4 Solving Multi-Step Equations
Consecutive Integers -
Order in which to solve multi step equations:
1.
2.
1.
2.
3.
Solve and check the following equations:
1.
7m – 17 = 60
2.
t
 21 14
8
4.
8 – 10x = -198

3.

p 15
 6
9
5.
27x – 176 = 121
6.
45 – 20y = -105
Writing and solving multi-step equations:
7. Two-thirds of a number minus 6 is -10.
8. The product of 36 and a number decreased by 15 is -123. Find the number.
9. The difference of 6 times a number and 8 is 34. Find the number.
Consecutive Integer Problems
10. Find 2 consecutive integers whose sum is 187.
11. Find 3 consecutive integers whose sum is -42.
12. Find 3 consecutive EVEN integers whose sum is 78.
2.5: Solving Equations with a Variable on Each Side
Ex. 1: Solve the following equations. Check your solutions.
1.) 2(2x + 4) = 16
2.) – 3(3x – 4) + 2x = 21 – 7
3.) 4(x – 3) + 3x – 7 = 2
4.) 2(4x – 3) – 3(x + 4) = 4
How to Solve Equations with a variable on both sides:
Steps:
1. Use the ______________________
to remove grouping symbols.
2. ___________ the expressions on each side of the equals sign
3. Use subtraction/ addition to get the _________ on the ____side
of the equal sign and the _________________________ on the
________side.
4. Combine _______________ (simplify) on each side
5.
Isolate the ______________ to solve
a. If the solution results in a ________ statement, there is _____ solution:
b. If the solution results in an __________, the solution is ______ numbers
Ex. 2: Simplify the expressions
5.) -2 + 10p = 8p – 1
7.) 4(2r  8) 
1
(49r  70)
7
6.) 2x + 4 = 6x – 12
8.) 2(3x – 8) = 10x + 4
Ex. 3: Special scenario. Solve the following equations. Check your solutions.
9.) 7 – 3r = r – 4(2 + r)
11.) 2m + 5 = 5(m – 7) – 3m
10.) 5h – 7 = 5(h – 2) + 3
12.) 3(k + 1) – 5 = 3k – 2
2.6 Ratios and Proportions
Ratio:
Proportion:
Extremes:
Means:
Determine if the following ratios form proportions.
1.
4
28
and
5
35

2.

20
30
and
24
38
3.

Means-Extremes Property of Proportions
*Words:
*Symbols:
*Examples:
Use the Means-Extremes Property to solve the following proportions.
56
32
and
63
36
4.
n 24

15 16

5.
10 34

15 x

7.
5.2 x

6.2 10

6.
21 28

m 8
*9.
x  2 18

8
24

8.

1 x

8 30

Applications
10.
Trent goes on a 30-mile bike ride every Saturday. He rides the distance in 4 hours. At
this rate, how far can he ride in 6 hours?
11.
The scale of a map for Crater Lake National Park is 2 inches = 9 miles. The distance
between Discovery Point and Phantom Ship Overlook on the map is about 1
the actual distance between these 2 places?

3
inches. What is
4
2.7 Percent of Change
change
r

original 100
State
 whether each percent of change is a percent of increase or a percent of decrease. Then
find each percent of change.
1. Original: 32
New: 40
2. Original: 20
New: 4
Find Amount After Sales Tax:
3. A meal for two at a restaurant costs $ 32.75. If the sales tax is 5%, what is the total
price of the meal?
Find Amount After Discount:
4. A dog toy is on sale for 20% off the original price. If the original price of the toy is $ 3.80,
what is the discounted price?
Percent Proportion:
is
%

of 100
5. 16 is what percent of 40?

6. Find 20% of 94.
7. 5% of what number is 4.5?
2.8 Solving for a Specific Variable
Solve for Variables: Some equations contain more than one variable. It is often useful
and asked by the directions to solve for one of these variables.
Directions: Solve for a Specific Variable
Example 1:
3x – 4y = 7
1A.
(Solve for y)
15 = 3n + 6p
(Solve for n)
1B.
𝑘−2
5
= 11𝑗
Example 2:
The formula for the circumference of a circle is = 2𝜋𝑟 , C represents the circumference
and r represents radius.
a. Solve for r
b. Find the radius, C = 32.7
2.9 Weighted Averages
Weighted Average: Is the sum of the product of a number of units and the value per
unit, divided by the sum of the number of units.
Mixture Problems: When two or more parts are combined into a whole, are solved using
weighted averages.
Example 1: How many pounds of mixed peanuts selling for $4.75 per pound should be
mixed with 10 pounds of dried fruit selling for $5.50 per pound to obtain a trail mix that
sells for $4.95 per pound?
*** Let w = the number of points of mixed peanuts. Fill in the table ***
Units (lb)
Dried Fruit
Mixed Peanuts
Trail Mix
Price per Unit (lb)
Total Price
Uniform Motion Problems: Are problems where an object moves at a
certain speed, or rate.
Formula: d = rt
d = Distance , r = Rate , t = Time
Example 2: A car and an emergency vehicle are heading toward each other. The car is
traveling at a speed of 30 miles per hour or about 44 feet per second. The emergency
vehicle is traveling at a speed of 50 miles per hour or about 74 feet per second. Under
ideal conditions a siren can be heard from up to 440 feet. The two vehicles in this
problem are 1000 feet apart. How many second will the driver hear the first siren?
*** Draw Picture ***
r
Car
Emergency Vehicle
t
d = rt