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
Solving for the Unknown: A how-to
Approach for Solving Equations
Learning Unit Objectives
LU5.1 Solving Equations for the Unknown
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
5-2
#5
Solving for the Unknown: A how-to
Approach for Solving Equations
Learning Unit Objectives
LU5.2 Solving Word Problems for the Unknown
1. List the steps for solving word problems
2. Complete blueprint aids to solve word
problems; check the solutions
5-3
Terminology
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.
5-4
Variables and constants are terms of mathematical
expressions.
Solving Equations for the Unknown
Equality in equations
A+8
Left side of equation
58
Right side of equation
Dick’s age in 8 years will equal 58
5-5
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
5-6
Solving for the Unknown Rule
Whatever you do to
one side of an
equation, you must do
to the other side.
5-7
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.
5-8
Opposite Process Rule
A + 8 = 58
- 8 - 8
A
Check
50 + 8 = 58
5-9
= 50
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.
5-10
Equation Equality Rule
7G = 35
7G = 35
7
7
G = 5
Check
7(5) = 35
5-11
Multiple Processes Rule
When solving for an
unknown that
involves more than
one process, do the
addition and
subtraction before the
multiplication and
division.
5-12
Multiple Process Rule
H +2= 5
4
H +2= 5
4
-2
-2
H = 3
4
Check
12 + 2 = 5
4
3 + 2= 5
5-13
(4)
( H4) = 4(3)
H = 12
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.
5-14
Parentheses Rule
5(P - 4) = 20
5P – 20 = 20
+20 +20
5P = 40
Check
5(8-4) = 20
5(4) = 20
20 = 20
5-15
5P = 40
5
5
P =8
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.
5-16
Like Unknown Rule
4A + A = 20
5A = 20
5A = 20
5
5
Check
4(4) +4 = 20
16 + 4 = 20
5-17
A=4
Solving Word Problems for Unknowns
1) Read the entire Problem
3) Let a variable
represent the unknown
Y = Computers
Read again if necessary
2) Ask: “What is the
problem looking for?”
4) Visualize the
relationship between
the unknowns and
variables. Then set up
an equation to solve
for unknown(s)
4Y + Y = 600
5) Check your results
to ensure accuracy
5-18
Solving Word Problems for the Unknown
Blueprint aid
Unknown(s)
5-19
Variable(s)
Relationship
Solving Word Problems for the Unknown
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?
Unknown(s)
Variable(s)
Cars Sold
ICM
Ring
4C
C
4C - C = 27
3C = 27
3
3
C=9
5-20
Relationship
4C
-C
27
Ring = 9 computers
ICM = 4(9) = 36 Computers
Check 36 - 9 = 27
Problem 5-34:
Solution:
4S + S = 5,500
5S = 5,500
5
5
S = 1,100
4S = 4,400
5-21
Unknown(s) Variable(s) Relationship
Shift 1
4S
4S (4,400)
Shift 2
S
+ S (1,100)
5,500
Problem 5-36:
Solution:
T + 3T = $40,000
4T = $40,000
4
4
T = $10,000
3T = $30,000
Unknown(s)
Variable(s)
Jim
T
Phyllis
3T
Relationship
T ($10,000)
+ 3T ($30,000)
$40,000
5-22
Problem 5-38:
Solution:
Check:
14B + 6B = 1,200
1,200
20B
=
20
20
B = 60 bottles
60($6) + 420($2) = $1,200
$360 + $840 = $1,200
$1,200 = $1,200
7B = 420 thermometers
Unknown(s)
Thermometers
Hot-water Bottles
Variable(s)
Price
Relationship
7B
$2
14B
B
6
+6B
Total = $1,200
5-23