Chapter 2 Notes­Pre­AICE I
2.1 Solving One­Step Equations
isolate: getting the variable with a coefficient of 1 and on one side of the equation
inverse operation: undo an operation (+/­) & (x/÷)
reciprocal: flip the fraction *multiply both sides by reciprocal if fraction is next to variable*
2.1 Solving One­Step Equations
Solve.
27 + n = 46
4 = q + 13
c + 4 = ­9
­5 + a = 21
5.5 = ­2 + d
67 = w ­ 65
­7y = 28
6b = 0.96
35 = j/5
k
2
­1 = 5/8x
/7 = 2
/3g= 12
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Chapter 2 Notes­Pre­AICE I
2.2 Solving Two­Step Equations
PEMDAS *with equations you go backwards*
Add/Subtract FIRST
Multiply/Divide SECOND
*multiply by denominator (bottom of fraction) on both sides to "clear" fraction from one side
2.2 Solving Two­Step Equations
Solve.
2 + a/4 = ­1
­1 = 7 + 8x
4b + 6 = ­2
7= x­8
3
h­3 5
2
2 = d+17 = 5 1/2
2
A train is 100 miles away. Its is approaching the station at 40 miles/hour. How long will it take to arrive?
A themepark charges $50 for entry plus $4 per ride. If someone spent $82, how many rides did they take?
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Chapter 2 Notes­Pre­AICE I
2.3 Solving Multi­Step Equations
Steps to Solve:
­distribute
­combine like terms on same side
­add or subtract
­multiply or divide
*multiply whole equation by LCM of all denominators to clear fractions*
2.3 Solving Multi­Step Equations
Solve:
6p ­ 2 ­ 3p =16
17 = p ­ 3 ­ 3p
64 = 8(r + 2)
7(f ­ 1) = 45
2m
/7 + 3m/14 = 1
1.06g ­ 3 = 0.71
/4 ­ x/3 = 10
3x
25.24 = 5g + 3.89
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Chapter 2 Notes­Pre­AICE I
2.4 Solving Equations w/ Variables on Both Sides
**What if the variables cancel out?**
­double check work (for sign errors)
­if answer is true (3=3, 0=0, etc.) it's called an identity (infinitely many solutions)
­if answer is false (3≠4, 0≠9, etc.) it has no solution.
identity­ an equation that's true for every possible value of the variable
2.4 Solving Equations w/ Variables on Both Sides
Solve.
3x + 4 = 5x ­ 10
5(y ­ 4) = 7(2y + 1)
2a + 3 = 1/2(6 + 4a)
4x ­ 5 = 2(2x + 1)
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Chapter 2 Notes­Pre­AICE I
2.5 Literal Equations
literal equation: equation with 2 or more variables
formula: an equation that represents a relationship among quantities
*Use inverse operations to isolate selected variable*
*SAME as normal equations* (treat other letters like numbers)
2.5 Literal Equations
Examples:
Solve each equation for y, then find y for the value of x.
1) y=2x + 5; x= ­1, 0, 3
2) 8x = ­4y + 4; x= 1, 2, 3
Solve each equation for x.
3) mx + nx = c
4) x­s =1
t
5) 5(x­b) = x
6) What is the radius of a circle with a circumference of 21?
7) A triangle has a height of 5 ft and an area of 25 ft2. What is the length of its base?
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Chapter 2 Notes­Pre­AICE I
2.6 Ratios, Rates, and Conversions
ratio­ compares two numbers by division
rate­ a ratio that compares different quantities measured in units
unit rate­ a rate with denominator of 1 (you divide to get this)
conversion factor­ the ratio of two equivalent measures in different units (example 1ft )
12 in
2.6 Examples
Convert the amount to the given unit:
b) 3 mi; yards
a) 330 min; hours
A cookie factory operates 15 hours a day for 20 days and makes 76, 500 cookies. How many cookies an hour does it make?
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Chapter 2 Notes­Pre­AICE I
2.7 Solving Proportions
proportions­ an equation that states two ratios are equal
cross products­ when you multiply diagonal with proportions (these are equal)
Examples:
Solve each proportion:
b) 3 = 1
a) 5 = 15
9 x x+1 2
d) x+2 c) p =7
= 3
4 8 2x­6 8
e) A florist is making centerpieces. He uses 2 dozen roses for every 5 centerpieces. How many dozens of roses will he need to make 20 centerpieces?
2.8 Proportions & Similar Figures
similar figures­ have the same shape, but not the same size
*the ratios of side lengths are equal*
scale­ the ratio of a small drawing or figure to the actual real life object or distance
Examples:
Each figure is similar. Find the missing length.
b)
a)
4
3
18
x
x
16
12
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The scale of a map is 1 cm: 15 km. Find the actual distance.
a) 2.5 cm
b) 15 cm
c) 4.6 cm
A student who is 5.25 ft tall is standing next to a tree. The shadow of the student is 20 ft and the shadow of the tree is 80 ft. What is the height of the tree?
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Chapter 2 Notes­Pre­AICE I
2.9 Percents
Key Words:
is
%
=
is =
of
100
of *
a number x
*change given percents to decimals*
Examples:
What percent of 75 is 15?
What is 25% of 144?
20% of what number is 80?
A tennis racket is usually $65. The racket is on sale for 20% off. What is the sale price?
Water covers 11,800 mi2 of Florida. This is 18% of the total area of Florida. What is the total area of Florida?
2.10 Percent of Change
p% = amount of increase/decrease
original
Examples:
Tell if percent of change is an increase or decrease. Then find the percent change. Round to the nearest percent.
Original: 12
New: 18
Original: 2008
New: 1975
An employee was hired at $8 per hour. After a raise, she makes $8.75. What is the percent increase?
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