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Algebra Chapter 2
Solving Linear Equations
2.1 Find Square Roots and Compare Real Numbers
Square Root: __________________________________________________________________________
Radicand: ____________________________________________________________________________
Perfect Square: ________________________________________________________________________
Irrational Number: _____________________________________________________________________
Real Numbers: ________________________________________________________________________
Example 1: Find the square roots.
 16
400
Example 2: Approximate
 81
52 to the nearest integer.
Example 3: Tell whether each of the following numbers is a real number, a rational number, an irrational number,
an integer, or a whole number: 64 17  36
Number
Real
number?
Rational
number?
Irrational
number?
Integer?
Whole
number?
64
17
 36
Example 4: Order the numbers from least to greatest:
3
, 16 ,2.2, 12 , 6
5
Example 5: Rewrite the given conditional statement in if-then form. Then tell whether the statement is true or
false. If false, give a counterexample.
a. Given: No square roots are rational numbers.
b. Given: All integers are rational numbers.
2.2 Solve One-Step Equations
Inverse operations: _____________________________________________________________________
Equivalent equations: ___________________________________________________________________
Reciprocal: ___________________________________________________________________________
Properties of Equality
Addition Property of Equality: ____________________________________________________________
Subtraction Property of Equality: _________________________________________________________
Multiplication Property of Equality: _______________________________________________________
Division Property of Equality: ____________________________________________________________
Example 1: Solve x + 11 = 15.
Example 2: Solve x – 8 = 17.
Example 3: Solve 7x = –63.
Example 4: Solve
Example 5: Solve
x
=4
12
3
x6
5
Example 6: In the 2004 Summer Olympics, Inge de Bruijn won the women’s 50-meter freestyle. Her winning
time was 24.58 seconds. Find her average swimming speed to the nearest hundredth of a meter per second.
2.3 Solve Two-Step Equations
Example 1: Solve 3x – 7 = 8.
Example 2: Solve 11x – 9x = 14.
Example 3: The output of a function is 7 more than twice the input. Find the input when the output is 13.
Example 4: To rent a booth at the county fairgrounds cost $42 per day plus a onetime equipment fee of $85. Find
the number of days Mr. Batzle rented a booth if he paid a total of $337. Write and solve an equation to find the
answer.
2.4 Solve Multi-Step Equations
Example 1: Solve 17x – 11x + 8 = 20.
Example 4: Solve
Example 2: Solve 4x + 3(2x – 1) = 17.
3
(5x – 4) = 12.
4
Example 5: Edmund and Roberto took a 7 day (168 hours), 90 mile canoe trip down the Allagash River. If they
paddled at an average rate of 2.5 miles per hour, how many hours did they not spend paddling? Write an equation
to find the answer.
2.5 Solve Equations with Variables on Both Sides
Identity: _____________________________________________________________________________
Example 1: Solve 13 – 6x = 3x – 14.
Example 2: Solve 4x – 7 =
3
(9x – 15)
4
Example 3: A music website sold 94 single songs and 67 albums today. The number of single downloads has
been increasing by 22 each day. The number of album downloads has been decreasing by 5 each day. If these
trends continue, in how many days will the number of single downloads will be ten times the number of album
downloads? Write and solve an equation to find the number of days.
Example 4: Solve the equation, if possible.
a. 4(3x – 2) = 2(6x + 1)
b. 4(4x – 5) = 2(8x – 10)
2.6 Write Ratios and Proportions
Ratio: _______________________________________________________________________________
Proportion: ___________________________________________________________________________
Simplest form: ________________________________________________________________________
Example 1: A shoe store sells 15 pairs of women’s shoes and 12 pairs of men’s shoes. Find the specified ratio.
a. Find the ratio of women’s shoe sales to men’s shoe sales.
b. Find the ratio of men’s shoe sales to total shoe sales.
Example 2: Solve a proportion
7
9
=
x
54
Example 3: A restaurant owner uses 3 cloves of garlic for every 5 pints of sauce. The restaurant uses 210 pints of
sauce during the day. Find the number of cloves of garlic the restaurant uses to make the sauce.
2.7 Solve Proportions Using Cross Products
Cross Product: ________________________________________________________________________
Scale Drawing: ________________________________________________________________________
Scale Model: _________________________________________________________________________
Scale: _______________________________________________________________________________
20 8
Example 1: Solve
=
35 x
Example 3: A bag of large breed dog food recommends feeding a dog 3 cups of food a day for every 40 pounds
of body weight. A dog weights 98 pounds. How much food should the dog be eating each day?
Example 4: A blueprint of an office building has a scale of 2 inches:15 feet. A completed scale model of the
building is about 14.5 inches tall. Estimate the actual height of the office building.
2.8 Rewrite Equations and Formulas
Literal Equation: ______________________________________________________________________
Formula: _____________________________________________________________________________
Example 1: Solve p + qx = r for x. Then use the solution to solve 3 + 5x = -7.
Example 2: Write 9x – 4y = 8 so that y is a function of x.
Example 3: The formula for the volume of a rectangular prism is V = lwh. Solve the formula for l.
Example 4: The area A of a triangle is given by the formula A =
1
bh where b is the base and h is the height.
2
a. Solve the formula for the base b.
b. Use the rewritten formula to find the base of the triangle shown, which has an area of 106.8 square inches.
Example 5: Irina deposited $650 in a savings account. After two years her account balance was $682.50. Find
the rate of interest for the two years. Use the formulas A =P(1 +rt), where A is the account balance, P is the
principal, r is the rate, and t is the time. Rewrite the formula to isolate r and then solve.