Chapter 3 Math
Notes, Examples, and Practice
Solving Equations Using Addition and Subtraction
Move 3 to other side of equal sign to find x
Numbers switch signs ( - or + ) when they cross the equal sign
Subtract 3 from 10
Follow rules of addition, but instead of subtracting, add to other side.
Add 5 to 4
The temperature rose 15° to -7°F. What was the original temperature.
Write equation for problem ⇾ x+15=-7
Solve equation ⇾ x=-22°F
Solving Equations Using Multiplication and Division
Multiply each side by the same nonzero number
X/2=3 X=6 (multiply by 2)
4X=12 X=3 (divide by 3)
You do the inverse operation in order to isolate X; If it is dividing, Multiply each side, and if it is multiplying, divide each side.
-4X=1 -write original equation
-4X/4= 1/-4 -divide each side by -4
X= -¼ -simplify
X/5= -30 - write original equation
5(X/5)= 5(-30) -multiply each side by 5X= -150 -simplify
When it is a fraction- multiply by the reciprocal
10= -2/3X -write original equation
(-3/2) 10= (-3/2)(-2/3X) -multiply each side by -3/2-15=m -simplify
Solving Multi-Step Equations
Simplify one or both sides of the equation (if needed)
- Use inverse operations to isolate variable
Example 1 - Solving Linear Equations
1/3x+6=-8 Write original equation
1/3x+6=-8 Subtract 6 from each side
3(1/3x)=3(-14) Multiply each side by 3x=-42 Simplify
Example 2 - Combining Like terms
7x-3x-8=24 Write original equation
4x-8=24 Combine like terms
4x-8=24 Add 8 to each side
Solving Equations with Variables on Both Sides
Doesn't matter which side variables are moved to, answer will be the same either way
Example 1 -
Solve 7x + 19 = -2x + 55
7x + 19 + 2x = -2x + 55 + 2x Add 2 to both sides
9x + 19 = 55 Simplify
9x + 19 - 19 = 55 - 19 Subtract 19 from both sides
9x = 36 Simplifyx=4
UPS charges $7 for the first pound, and $0.20 for each additional pound. FedEx charges $5 for the first pound and $0.30 for each additional pound. How many pounds, p, will it take for UPS and FedEx to cost the same?
7+0.2p = 5+0.3p
7+0.2p-0.2p = 5+0.3p-0.2p Subtract 0.2 from both sides
7 = 5+0.1p Simplify
7-5 = 5+0.1p-5 Subtract 5 from both sides
2 = 0.1p Simplify20 = p
Linear Equations and Problem Solving
Example -At East High School., 579 students take spanish. This number has been increasing at a rate of about 30 students per year. The number of students taking French is 217 and has been decreasing at a rate of about 2 students per year. At these rates, when will there be three times as many students taking Spanish as taking French.
# taking Spanish now + Rate of increase for Spanish x # of years = 3 ( # taking Spanish now - rate of decrease for French x # of years )
Solving Decimal Equations
Three people want to share equally in the cost of a pizza. The pizza costs $12.89.
You can find what each person will have to pay by solving 3X= 12.89
3X= 12.89 -write original equation
X= 4.2966… -the exact answer is a repeating variable
X = 4.30 -round to the nearest cent.
Formulas and Functions
Function- a relationship or expression involving one or more variables.
Example 1 -
A=lw Solve for length l.
A/w=l To get l by itself, divide each side by w
(Substitute given values into new formula)
The length of rectangle is 5 feet.
Example 2 -
C=5/9(F - 32) Solve for F
9/5 ⋅ C = 9/5 ⋅ 5/9(F - 32) Multiply each side by 9/5
9/5C = F - 32 Simplify
9/5C + 32 = F - 32 + 32 Add 32 to each side
9/5C + 32 = F
Rates, Ratios, and Percents
60 miles / 3 hours Simplify
20 miles / 1 hour
Ratio- the quantitative relation between two amounts showing the number of times one value contains or is contained within the other
9:3 or 9 to 3 or 9/3 Simplify
3:1 , 3 to 1, 3/1
Percent- one part in every hundred
60% of 200 Simplify
30% of 100