Chapter FourSOLVING FOR

THE UNKNOWN

Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

LU 4-1: Solving Equations for the Unknown

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

2.List the fi ve 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

3.List the steps for solving word problems.

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

LEARNING UNIT OBJECTIVES

4-2

TERMINOLOGY

4-3

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.

VARIABLES AND CONSTANTS RULES

4-4

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

SOLVING EQUATIONS FOR THE UNKNOWN

4-5

Left side of equation Right side of equation

Equality in equations

A + 8 58

Dick’s age in 8 years will equal 58.

SOLVING FOR THE UNKNOWN RULE

4-6

Whatever you do to one side of an equation, you

must do to the other side.

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

3 + 5(1+2³)² 1. Simplify inside theparentheses by applyingthe exponent: 2³ = 8 and8 + 1 = 9.

2. Apply the exponent: (9)² = 81.3. Multiply: 5(81) x 405.4. Add: 3 + 405 = 408.

3 + 5(9)² 3 + 5(81) 3 + 405 = 408

Example

When we apply theOrder of Operations,the answer is 408.

Mechanical steps3 + 5(1+2³)²

Explanation

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

When you combine like terms, your equation is simplified and easier to solve.

Mechanical steps

9x + 4 = 30 – 4x

Explanation

Combine like terms on each side of the equation.

4-9

OPPOSITE PROCESS RULE

4-10

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.

EQUATION EQUALITY RULE

4-11

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.

DRILL 1: SUBTRACTING SAME NUMBER FROM BOTH SIDES OF

EQUATION

4-12

A + 8 = 58

- 8 - 8

A = 50

Check

50 + 8 = 58

8 is subtracted from both sides of equation to isolate variable A on the left.

DRILL 2: ADDING SAME NUMBER TO BOTH SIDES OF EQUATION

Example Mechanical steps Explanation

B - 50 = 80 +50 +50 B = 130

B - 50 = 80

Some number B less 50 equals 80.

50 is added to both sides to isolate variable B on the left.

Check130 - 50 = 80 80 = 80

4-13

DRILL 3: DIVIDING BOTH SIDES OF EQUATION BY SAME NUMBER

4-14

7G = 35

7G = 357 7

G = 5

Check

7(5) = 35

By dividing both sides by 7, G equals 5.

DRILL 4: MULTIPLYING BOTH SIDES OF EQUATION BY SAME NUMBER

Example Mechanical steps Explanation

V5

= 70

Some number Vdivided by 5equals 70.

V5

= 70

(5)V5

= 705

V = 350

By multiplying both sides by5, V is equal to 350.

Check350

5= 70

70 = 70

4-15

MULTIPLE PROCESSES RULE

4-16

When solving for an unknown that involves more than one process, do the addition and subtraction before the multiplication and division.

MULTIPLE PROCESS RULE

4-17

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

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.

PARENTHESES RULE

4-18

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.

PARENTHESES RULE

4-19

5(P -- 4) = 20

5P – 20 = 20

+ 20 + 20

Check

5(8 - 4) = 20

5(4) = 20

20 = 20

1. Parentheses tell us that everything inside parentheses is multiplied by 5.

Multiply 5 by P and 5 by 4. 5P = 40

5P = 40 5 5 P = 8

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.

Explanation

LIKE UNKNOWN RULE

4-20

To solve equations with like unknowns, you first combine the unknowns and then solve with the opposite process used in the equation.

LIKE UNKNOWN RULE

4-21

4A + A = 20

5A = 20

5A = 205 5

A = 4

Check

4(4) +4 = 20 16 + 4 = 20

To solve this equation: 4A + 1A = 5A. Thus, 5A = 20. To solve for A, divide both sides by 5, leaving A equals 4.

Explanation

SOLVING WORD PROBLEMS FOR UNKNOWNS

4-22

3) Let a variable represent the unknown.Y = Computers

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.

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

1) Read the entire problem.

Read again if necessary!

SOLVING WORD PROBLEMS FOR THE UNKNOWN

4-23

Unknown(s) Variable(s) Relationship

Blueprint aid

SOLVING WORD PROBLEMS FOR THE UNKNOWN

4-24

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 3 C = 9

Ring = 9 computers

ICM = 4(9) = 36 computers

Cars Sold:ICM Ring

Check 36 -- 9 = 27

4C C

4C-- C 27

Mechanical Steps

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

4-25

Men ――――― Women

2.

Men : Women3.

RATIOS AND PROPORTIONS

A proportion is made up of two ratios whose cross products are equal.

4-26

5――15

1 ―― 3

‗(5 x 3 = 15)

(15 x 1 = 15)

X――15

1 ―― 3

‗ 3X = 15 3 3

X = 5