SOLVING FOR THE
UNKNOWN
McGraw-Hill/Irwin
Chapter Four
Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.
LEARNING UNIT OBJECTIVES
LU 4-1: 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.
LU 4-2: 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.
4-2
TERMINOLOGY
Expression – A meaningful combination of numbers and letters
called terms.
Equation – A mathematical statement with an equals 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.
4-3
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
4-4
SOLVING EQUATIONS FOR THE
UNKNOWN
Equality in equations
A+8
58
Left side of equation
Right side of equation
Dick’s age in 8 years will equal 58.
4-5
SOLVING FOR THE UNKNOWN RULE
Whatever you do to one side of
an equation, you must do to the
other side.
4-6
ORDER OF OPERATIONS
1. Simplify inside parentheses or brackets.
2. Apply exponents.
3. Multiply or divide from left to right.
4. Add or subtract from left to right.
4-7
ORDER OF OPERATIONS
Example
3 + 5(1+2³)²
Mechanical steps
3 + 5(1+2³)²
When we apply the
Order of Operations,
the answer is 408.
Explanation
1. Simplify inside the
parentheses by applying
the exponent: 2³ = 8
and
8 + 1 = 9.
2. Apply the exponent:
3 + 5(9)²
3 + 5(81)
3 + 405 = 408
(9)² = 81.
3. Multiply: 5(81) x 405.
4. Add: 3 + 405 = 408.
4-8
COMBINING LIKE TERMS
Add or subtract the coefficients
of like variables to simplify an
expression or equation.
Example
8x + 6 – 2 + x = 20 - 4x + 10
Mechanical steps
9x + 4 = 30 – 4x
Explanation
Combine like terms on each
side of the equation.
When you combine like
terms, your equation is
simplified and easier to
solve.
4-9
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. For
example, if the equation process is addition, solve for the
unknown by using subtraction.
If +, then -.
If -, then +.
If x, then ÷.
If ÷, then x.
4-10
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 zero) without affecting the equality of
the equation.
4-11
DRILL 1: SUBTRACTING SAME NUMBER
FROM BOTH SIDES OF EQUATION
A + 8 = 58
-
A
8
- 8
= 50
8 is subtracted from
both sides of equation to
isolate variable A on the
left.
Check
50 + 8 = 58
4-12
DRILL 2: ADDING SAME NUMBER TO
BOTH SIDES OF EQUATION
Example
B - 50 = 80
Some number B less
50 equals 80.
Mechanical steps
B - 50 = 80
+50 +50
B = 130
Explanation
50 is added to both sides to
isolate variable B on the
left.
Check
130 - 50 = 80
80 = 80
4-13
DRILL 3: DIVIDING BOTH SIDES OF
EQUATION BY SAME NUMBER
7G = 35
7G = 35
7
7
By dividing both sides
by 7, G equals 5.
G = 5
Check
7(5) = 35
4-14
DRILL 4: MULTIPLYING BOTH SIDES OF
EQUATION BY SAME NUMBER
Example
V
=
5
Mechanical steps
V
=
5
70
Some number V
divided by 5
equals 70.
5
V
=
5
70
Explanation
By multiplying both
sides by
5, V is equal to 350.
70 (5)
V = 350
Check
350 = 70
5
70 = 70
4-15
MULTIPLE PROCESSES RULE
When solving for an unknown
that involves more than one
process, do the addition and
subtraction before the
multiplication and division.
4-16
MULTIPLE PROCESS RULE
H +2= 5
4
H +2= 5
4
-2 -2
4
H
4
= 3
H
4
= 4(3)
H
=
()
12
Step 1. Move constant to right
side by subtracting 2 from
both sides.
Step 2. To isolate H, which is
divided by 4, we do the
opposite process and
multiply 4 times both
sides of the equation.
Check
12 + 2 = 5
4
3+2=5
4-17
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. This is known as the distributive
rule.
Then you continue to solve for the unknown with
the opposite process used in the equation. Do
the addition and subtraction first; then do the
multiplication and division.
4-18
PARENTHESES RULE
5(P -- 4) = 20
5P – 20 = 20
+ 20 + 20
5P
= 40
5P
5
= 40
5
P
=
8
Explanation
1. Parentheses tell us that everything
inside parentheses is multiplied by 5.
Multiply 5 by P and 5 by 4.
2. Add 20 to both sides to isolate 5P
on left.
3. To remove 5 in front of P, divide
both sides by 5 to result in P
equals 8.
Check
5(8 - 4) = 20
5(4)
= 20
20
= 20
4-19
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.
4-20
LIKE UNKNOWN RULE
4A + A = 20
5A
= 20
5A
5
= 20
5
Explanation
To solve this equation:
4A + 1A = 5A. Thus, 5A
= 20. To solve for A,
divide both sides by 5,
leaving A equals 4.
A=4
Check
4(4) +4 = 20
16 + 4 = 20
4-21
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 the
unknown(s).
4Y + Y = 600
5) Check your results to
ensure accuracy.
4-22
SOLVING WORD PROBLEMS FOR THE
UNKNOWN
Blueprint aid
Unknown(s)
Variable(s)
Relationship
4-23
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)
Cars Sold:
ICM
Ring
Mechanical
Steps
4C -- C = 27
3C
= 27
3
3
C=9
Variable(s)
4C
C
Relationship
4C
-- C
27
Ring = 9 computers
ICM = 4(9) = 36 computers
Check 36 -- 9 = 27
4-24
RATIOS AND PROPORTIONS
A ratio is a fraction used for comparing two quantities.
Ratios and proportions are used in business to demonstrate
a relationship between two values.
Make note of the order of items in the fraction as it is
important to maintain that order.
1. Men to Women
2.
Men
―――――
Women
3. Men : Women
4-25
RATIOS AND PROPORTIONS
A proportion is made up of two ratios whose cross products are equal.
5
――
15
‗
1
――
3
X
――
15
‗
1
――
3
(5 x 3 = 15)
(15 x 1 = 15)
3X = 15
3
3
X=5
4-26