Chapter 05

Solving for the Unknown: A

How-To Approach for

Solving 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 Learning Unit Objectives Solving Equations for the Unknown LU5.1

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 Learning Unit Objectives Solving Word Problems for the Unknown LU5.2

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 = 35 7 7

G = 5

Check

7(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) Check

12 + 2 = 5

4

3 + 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 = 40 5 5

P =8

Check

5(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 = 20 5 5

A = 4 Check

4(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 = 27 3C = 27

3 3 C = 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) Relationship

Shift 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