Chapter 2

Algebra

Objectives

Solve linear equations Solve mixture problems Solve rational equations Perform formulae manipulation Evaluate problems using ratios and

percents Solve percent problems

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Objectives (cont’d.)

Use the properties of exponents Use scientific notation Evaluate significant digits Use the scientific calculator to evaluate

expressions

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Solving Linear Equations

If the product of two numbers is 1, they are reciprocals• The reciprocal of 1 ⁄ 7 is 7

– .

Like terms have the same variable and the same exponent• Can be combined: 5x + 3x = 8x

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Solving Linear Equations (cont’d.)

Whatever operation is performed on one side must also be done to the other side

When solving any equation, the goal is to isolate the variable• Solve: 2x − 6 = 20

– Add 6 to both sides– Divide both sides by 2– Simplify: x = 13

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Solving Linear Equations (cont’d.)

Distributive property: • a(b + c) = ab + ac

Commutative property: • a + b = b + a• a × b = b × a

Associative property:• (a + b) + c = a + (b + c)• (a × b) × c = a × (b × c)

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Mixture Problems

A 3% solution is needed• Only 30 fl oz of a 4% solution is in stock• How much “neutral” solution should be added

to 30 fl oz of the 4% solution?x + 30

0%(x) + 4%(30) = 3%(x + 30)

0.00(x) + 0.04(30) = 0.03(x + 30)

1.2 = 0.03x + 0.9

0.03x = 0.3

x = 10

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Rational Equations

Equation containing rational expressions• Example:

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Formulae Manipulation

Sometimes we need work with formulas that do not have many numbers• Solve for A:

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.

Ratios and Proportions

Ratios can be written three ways:• 1 to 2 • ½ • 1:2

Ratios are in proportion if they are equivalent to each other:• 2/3 is proportional to 8/12

– .

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How to Calculate: Ratios and Proportions

Cross multiplication: When solving proportions:

• Components on left-hand side must be set up in the same order as components on right-hand side:

– .

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Solving Percent Problems (cont’d.)

Percents should be written as decimals• 35% of what number is 21?

.35 × x = 21

.35x = 21

21 ÷ 0.35

x = 60

Proportional formula:• When using this method do not use decimals

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Properties of Exponents

Product rule: exponentials are used to represent repeated multiplication• .

Quotient rule:• .• .

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Properties of Exponents (cont’d.)

Power rule for fractions:• .

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Properties of Exponents (cont’d.)

Negative exponent rule:• .

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Properties of Exponents (cont’d.)

Negative exponent rule for fractions:• .

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Properties of Exponents (cont’d.)

There is no exponent rule for adding exponentials• .

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Scientific Notation

Used when dealing with very large or very small numbers• .

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How to Calculate: Significant Digits

Significant digits tell about the accuracy of a measurement• Rule 1: Determining whether a digit is

significant:– All nonzero digits are significant– Zeros are significant if they are on the right side of

a decimal number– Zeros are significant if they are between two

significant digits

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How to Calculate: Significant Digits (cont’d.)

Rule 2: Determining whether a zero is not significant:• A zero is not significant if it is on the right side

of a whole number• A zero is not significant if it is on the left side

of a number

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Using the Scientific Calculator

When using the scientific calculator, keep order of operations in mind• PEMDAS• Key is used to enter expressions that contain

exponents: • To enter a negative number, enter the number

first and then enter the +⁄− key

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Using the Scientific Calculator (cont’d.)

.• .

.

• . .

• .

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Using the Scientific Calculator (cont’d.)

.• .

.• .

.• .

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If the product of two numbers is 1, the numbers are reciprocals

When solving an equation, the goal is to isolate or get the variable (x) by itself

When setting up proportions, components on both sides of equal sign must be set up the same

Percent problems can be solved by setting up an equation or by using a proportion

Summary

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The six rules for exponents are:• Product rule• Quotient rule• Power rule• Negative exponent rule• Exponent rules for fractions

Summary (cont’d.)