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 © 2010 Delmar, Cengage Learning. 2

Objectives (cont’d.)  Use the properties of exponents  Use scientific notation  Evaluate significant digits  Use the scientific calculator to evaluate expressions © 2010 Delmar, Cengage Learning. 3

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 © 2010 Delmar, Cengage Learning. 4

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 © 2010 Delmar, Cengage Learning. 5

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) © 2010 Delmar, Cengage Learning. 6

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 %(x) + 4%(30) = 3%(x + 30) 0.00(x) (30) = 0.03(x + 30) 1.2 = 0.03x x = 0.3 x = 10 © 2010 Delmar, Cengage Learning. 7

Rational Equations  Equation containing rational expressions Example: © 2010 Delmar, Cengage Learning. 8

Formulae Manipulation  Sometimes we need work with formulas that do not have many numbers Solve for A: © 2010 Delmar, Cengage Learning. 9.

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 –. © 2010 Delmar, Cengage Learning. 10

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: –. © 2010 Delmar, Cengage Learning. 11

Solving Percent Problems (cont’d.)  Percents should be written as decimals 35% of what number is 21?.35 × x = 21.35x = ÷ 0.35 x = 60  Proportional formula: When using this method do not use decimals © 2010 Delmar, Cengage Learning. 12

Properties of Exponents  Product rule: exponentials are used to represent repeated multiplication.  Quotient rule:. © 2010 Delmar, Cengage Learning. 13

Properties of Exponents (cont’d.)  Power rule for fractions:. © 2010 Delmar, Cengage Learning. 14

Properties of Exponents (cont’d.)  Negative exponent rule:. © 2010 Delmar, Cengage Learning. 15

Properties of Exponents (cont’d.)  Negative exponent rule for fractions:. © 2010 Delmar, Cengage Learning. 16

Properties of Exponents (cont’d.)  There is no exponent rule for adding exponentials. © 2010 Delmar, Cengage Learning. 17

Scientific Notation  Used when dealing with very large or very small numbers. © 2010 Delmar, Cengage Learning. 18

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 © 2010 Delmar, Cengage Learning. 19

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 © 2010 Delmar, Cengage Learning. 20

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 © 2010 Delmar, Cengage Learning. 21

Using the Scientific Calculator (cont’d.) .. .. .. © 2010 Delmar, Cengage Learning. 22

Using the Scientific Calculator (cont’d.) .. .. .. © 2010 Delmar, Cengage Learning. 23

© 2010 Delmar, Cengage Learning. 24  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

© 2010 Delmar, Cengage Learning. 25  The six rules for exponents are: Product rule Quotient rule Power rule Negative exponent rule Exponent rules for fractions Summary (cont’d.)