Chapter 05

Solving for the Unknown: A Solving for the Unknown: A How-To Approach for How-To Approach for

Solving EquationsSolving Equations

McGraw-Hill/Irwin Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.

5-2

1. Explain the basic procedures used to solve equations for the unknown

2. List the five rules and the mechanical steps used to solve for the unknown in seven situations; know how to check the answers

Solving for the Unknown: A how-to Approach for Solving Equations#5#5Learning Unit ObjectivesSolving Equations for the UnknownLU5.1LU5.1

5-3

1. List the steps for solving word problems

2. Complete blueprint aids to solve word problems; check the solutions

Solving for the Unknown: A how-to Approach for Solving Equations#5#5Learning Unit ObjectivesSolving Word Problems for the UnknownLU5.2LU5.2

5-4

Expression – A meaningful combination of numbers and letters called terms.

Equation – A mathematical statement with an equal sign showing that a mathematical expression on the left equals the mathematical expression on the right.

Formula – An equation that expresses in symbols a general fact, rule, or principle.

Variables and constants are terms of mathematical expressions.

Terminology

5-5

Solving Equations for the Unknown

Left side of equation Right side of equation

Equality in equations

A + 8 58

Dick’s age in 8 years will equal 58

5-6

Variables and Constants Rules

1. If no number is in front of a letter, it is a

1: B = 1B; C = 1C

2. If no sign is in front of a letter or

number, it is a +: C = +C; 4 = +4

5-7

Solving for the Unknown Rule

Whatever you do to one side of an

equation, you must do to the other side.

5-8

Opposite Process Rule

If an equation indicates a process such as addition,

subtraction, multiplication, or

division, solve for the unknown or variable by using the opposite

process.

5-9

Opposite Process Rule

A + 8 = 58

- 8 - 8

A = 50

Check

50 + 8 = 58

5-10

Equation Equality Rule

You can add the same quantity or number to both

sides of the equation and subtract the same quantity or number from both sides of the equation without affecting the equality of the equation. You

can also divide or multiply both sides of the equation by the same quantity or number (except 0) without affecting the equality of the equation.

5-11

Equation Equality Rule

7G = 35

7G = 357 7

G = 5

Check7(5) = 35

5-12

Multiple Processes Rule

When solving for an unknown that

involves more than one process, do the

addition and subtraction before the

multiplication and division.

5-13

Multiple Process Rule

H + 2 = 5 4

H + 2 = 5 4

-2 -2

H = 3 4

H = 4(3) 4

H = 12

( )(4)Check12 + 2 = 5 43 + 2= 5

5-14

Parentheses Rule

When equations contain parentheses (which indicates

grouping together, you solve for the unknown by first

multiplying each item inside the parentheses by the number or

letter just outside the parentheses. Then you continue

to solve for the unknown with the opposite process used in the equation. Do the addition and

subtractions first; then the multiplication and division.

5-15

Parentheses Rule

5(P - 4) = 20

5P – 20 = 20

+20 +20

5P = 40

5P = 405 5

P =8

Check5(8-4) = 20 5(4) = 20 20 = 20

5-16

Like Unknown Rule

To solve equations with like unknowns, you first combine the unknowns and then

solve with the opposite process used

in the equation.

5-17

Like Unknown Rule

4A + A = 20

5A = 20

5A = 205 5

A = 4Check4(4) +4 = 20 16 + 4 = 20

5-18

Solving Word Problems for Unknowns

1) Read the entire Problem

2) Ask: “What is the problem looking for?”

3) Let a variable represent the unknown

4) Visualize the relationship between the unknowns and variables. Then set up an equation to solve for unknown(s)

5) Check your results to ensure accuracy

Y = Computers

4Y + Y = 600

Read again if necessary

5-19

Solving Word Problems for the Unknown

Unknown(s) Variable(s) Relationship

Blueprint aid

5-20

Solving Word Problems for the Unknown

Unknown(s) Variable(s) Relationship

ICM Company sold 4 times as many computers as Ring Company. The difference in their sales is 27. How many computers of each company were sold?

4C - C = 273C = 27

3 3C = 9

Ring = 9 computers

ICM = 4(9) = 36 Computers

Cars Sold

ICM 4C 4C

Ring C -C

27

Check 36 - 9 = 27

5-21

Problem 5-34:

Solution:

Unknown(s) Variable(s) RelationshipShift 1 4S 4S (4,400) Shift 2 S + S (1,100) 5,500

4S + S = 5,500

= 5,500 5

S = 1,100

4S = 4,400

5S 5

5-22

Problem 5-36:

Solution:

Unknown(s) Variable(s) Relationship

Jim T T ($10,000)

Phyllis 3T + 3T ($30,000)

$40,000

T + 3T = $40,000

= $40,000

4

T = $10,000

3T = $30,000

4T 4

5-23

Problem 5-38:

Solution:

Unknown(s) Variable(s) Price Relationship

Thermometers 7B $2 14B

Hot-water Bottles B 6 +6B

Total = $1,200

14B + 6B = 1,200

=

B = 60 bottles

7B = 420 thermometers

20B 20

1,200 20

Check:60($6) + 420($2) = $1,200

$360 + $840 = $1,200

$1,200 = $1,200