Download Chapter 2 – Solving Linear Equations 2.1 – Writing Equations

Chapter 2 – Solving Linear Equations
2.1 – Writing Equations
Example: Translate each sentence into an equation.
A number b divided by three is six less than c.
Fifteen more than z times 6 is y times 2 minus eleven.
Problem-Solving Plan
Example: A jelly bean manufacturer produces 1,250,000 jelly beans per hour. How many hours does it take to
produce 10,000,000 jelly beans?
Formula
Example: Translate the sentence into a formula.
The perimeter of a square equals four times the length of the side.
In a right triangle, the square of the measure of the hypotenuse c is equal to the sum of the square of the
measures of the legs, a and b.
Example: Translate each equation into a verbal sentence.
12 − 2 = −5
+3 =
2.2 – Solving Equations by Using Addition and Subtraction
Addition Property of Equality:
Subtraction property of Equality:
Equivalent Equations:
To solve an equation:
Example: Solve each equation. Check your solution.
ℎ − 12 = −27
+ 63 = 92
+ 102 = 36
27 +
32 = − 8
= 30
+ =
7 = 42 +
Example: Write an equation for the problem. Then solve the equation.
Fourteen more than a number is equal to twenty-seven. Find the number.
Twenty-five is 3 less than a number. Find the number.
Example: The Washington Monument in Washington, DC, was built in two phases. During the first phase, the
monument was built to a height of 152 feet. During the second phase, additional construction resulted in the
monument’s final height of 555 feet. How much of the monument was added during the second construction
phase? Write an equation to solve the problem.
Chapter 2.3 – Solving Equations by Using Multiplication and Division
Multiplication Property of Equality:
Division Property of Equality:
Example: Solve each equation. Check your solution.
=3
11
= 143
−8 = 96
=−
84 = 3
−1
=
−42 = −3
=
Example: Write an equation for the problem. Then solve the equation
Negative fourteen times a number equals 224.
Chapter 2.4: Solving Multi-Step Equations
Solving Multi-Step Equations:
Example: Solve each equation. Check your solution.
5 − 13 = 37
− 9 = −11
+ 21 = 14
=4
8=3 +7
Example: Susan had a $10 coupon for the purchase of any item. She bought a coat that was on sale for ½ its original
price. After using the coupon, Susan paid $125 for the coat before taxes. What was the original price of the coat?
Write an equation for the problem. Then solve the equation.
Consecutive Integers:
Number Theory:
Example: Write an equation for the problem below. Then solve the equation and answer the problem.
Find three consecutive odd integers whose sum is 57.
Example: Write an equation for the problem below. Then solve the equation and answer the problem.
Find three consecutive integers whose sum is 21.
Chapter 2.5: Solving Equations with the Variable on Each Side
To solve an equation with variables on both sides:
Example: Solve and check your solution.
8+5 = 7 −2
2
+1=
1
−6
4
1
(18 + 12 ) = 6(2 − 7)
3
7( − 1) = −2(3 + )
Special Cases:
Example: Solve and check your solution.
8(5 − 2) = 10(32 + 4 )
7 + 5( − 1) = −5 + 12
1
4( + 20) = (20 + 400)
5
6( − 5) = 2(10 + 3 )
Example: Find the value of H so that the figures have the same area.
Chapter 2.6: Ratios and Proportions
Ratio:
Proportion:
Example: Determine whether the ratios and
Example: Determine whether the ratios
form a proportion.
and form a proportion.
Means:
Extremes:
Means-Extremes Property:
Example: Use cross products to determine whether each pair of ratios forms a proportion.
.
.
,
.
,
,
Solving Proportions:
Example: Solve the following proportions
=
8
=
25
40
3.2 2.6
=
4
Rate
Example: The gear on a bicycle is 8:5. This means that for every 8 turns of the pedals, the wheel turns 5 times.
Suppose the bicycle wheel turns about 2435 times during a trip. How many times would you have to crank the
pedals during the trip?
Scale
Example: In a road atlas, the scale for the map of Connecticut is 5 inches = 41 miles. The scale for the map of Texas
is 5 inches = 144 miles. What are the distances in miles represented by 2 inches on each map?
Chapter 2.7: Percent of Change
Percent of change:
Percent Proportion:
Percent Equation:
Example: State whether each percent of change is a percent of increase or decrease. Then find each percent of
change.
Original: 32
New: 40
Original: 20
New: 4
Original: 66
New: 30
Original: 9.8
New: 12.1
Example: The price a used-book store pays to buy a book is $5. The store sells the book for 28% above the price that
it pays for the book. What is the selling price of the book?
Example: A recent percent of increase in tuition at Northwestern University was 5.4%. If the new cost is $29,940 per
year, find the original cost per year.
Example: At one store the price of a pair of jeans is $26.00. At another store the same pair of jeans has a price that
is 22% higher. What is the price of jeans at the second store?
Example: A meal for two at a restaurant costs $32.75. if the sales tax is 5%, what is the total price of the meal?
Example: A new DVD costs $24.99. If the sales tax is 7.25%, what is the total cost?
Example: 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?
Example: A picture frame originally priced at $14.89 is on sale for 40% off. What is the discounted price?
Chapter 2.8: Solving for a Specific Variable
Example: Solve
5b + 12c = 9 for b.
7x – 2z = 4 – xy for x.
15 = 3n + 6p for n
d + 5c = 3d – 1 for d
= 11 for k
6q – 18 = qr + s for q
Example: A car’s fuel economy E (miles per gallon) is given by the formula E = m/g, where m is the number of miles
and g is the number of gallons of fuel used.
Solve the formula for m.
If Claudia’s car has an average fuel consumption of 30 miles per gallon and she used 9.5 gallons, how far did
she drive?
Example: The formula for the volume of a rectangular prism is V = lwh, where l is the length, w is the width, and h is
the height.
Solve the formula for w.
Find the width of a rectangular prism that has a volume of 79.04 cubic centimeters, a length of 5.2
centimeters, and a height of 4 centimeters.
Dimensional Analysis
Example: The formula for the volume of a cylinder is V = πr2h, where r is the radius of the cylinder and h is the
height.
Solve the formula for h.
What is the height of a cylindrical swimming pool that has a radius of 12 feet and a volume of 1810 cubic
feet?