1. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 1 Section 1.1 Introduction to Algebra: Variables and Mathematical Models

2. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 2 Variables in Algebra Algebra uses letters such as x and y to represent numbers. If a letter is used to represent various numbers, it is called a variable. For example, the variable x might represent the number of minutes you can lie in the sun without burning when you are not wearing sunscreen.

3. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 3 Variables in Algebra Suppose you are wearing number 6 sunscreen. If you can normally lie in the sun x minutes without burning, with the number 6 sunscreen, you can lie in the sun 6 times as long without burning - that is, 6 times x or 6x would represent your exposure time without burning.

4. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 4 Algebraic Expressions A combination of variables and numbers using the operations of addition, subtraction, multiplication, or division, as well as powers or roots, is called an algebraic expression.

5. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 5 Objective 1 Evaluate algebraic expressions.

6. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 6 Order of Operations - PEMDAS Order of Operations 1.Perform all operations within grouping symbols, such as parentheses. 2.Do all multiplications in the order in which they occur from left to right. 3.Do all additions and subtractions in the order in which they occur from left to right.

7. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 7 Example Evaluate each algebraic expression for

8. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 8 Objective 1: Example 1a. Evaluate the expression

9. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 9 Objective 1: Example 1b. Evaluate the expression

10. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 10 Objective 2 Translate English phrases into algebraic expressions.

11. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 11 Translating Phrases into Expressions English PhraseMathematical Operation sum plus increased by more than Addition difference minus decreased by less than Subtraction

12. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 12 Translating Phrases into Expressions English PhraseMathematical Operation product times of (used with fractions) twice Multiplication quotient divide per ratio Division

13. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 13 Objective 2: Example Write each English phrase as an algebraic expression. Let the variable x represent the number.

14. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 14 Objective 2: Example (cont)

15. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 15 Objective 3 Determine whether a number is a solution of an equation.

16. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 16 Equations An equation is a statement that two algebraic expressions are equal. An equation always contains the equality Some examples of equations are:

17. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 17 Equations Solutions of equations are values of the variable that make the equation a true statement. To determine whether a number is a solution, substitute that number for the variable and evaluate both sides of the equation. If the values on both sides of the equation are the same, the number is a solution. For example, 2 is a solution of x 4 3x since when we substitute the 2 for x, we get 2 4 3(2) or equivalently, 6 6

18. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 18 Objective 3: Example 3a. Determine whether the given number is a solution of the equation.

19. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 19 Objective 3: Example 3b. Determine whether the given number is a solution of the equation. 3 is a solution.

20. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 20 Objective 4 Translate English sentences into algebraic equations.

21. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 21 Translate English sentences into Equations Earlier in the section, we translated English phrases into algebraic expressions. Now we will translate English sentences into equations. You’ll find that there are a number of different words and phrases for an equation’s equality symbol. equals gives yields is the same as is/was/will be

22. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 22 Objective 4: Example 4a.Write the sentence as an equation. Let the variable x represent the number. The quotient of a number and 6 is 5.

23. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 23 Objective 4: Example 4b.Write the sentence as an equation. Let the variable x represent the number. Seven decreased by twice a number yields 1.

24. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 24 Objective 5 Evaluate formulas.

25. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 25 Formulas and Mathematical Models One aim of algebra is to provide a compact, symbolic description of the world. A formula is an equation that expresses a relationship between two or more variables.

26. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 26 Formulas and Mathematical Models One variety of crickets chirps faster as the temperature rises. You can calculate the temperature by applying the following formula: If you are sitting on your porch and hear 50 chirps per minute, then the temperature is:

27. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 27 Formulas and Mathematical Models The process of finding formulas to describe real-world phenomena is called mathematical modeling. Formulas together with the meaning assigned to the variables are called mathematical models.

28. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 28 Formulas and Mathematical Models In creating mathematical models, we strive for both simplicity and accuracy. For example, the cricket formula is relatively easy to use. But you should not get upset if you count 50 chirps per minute and the temperature is 53 degrees rather than 55. Many mathematical formulas give an approximate rather than exact description of the relationship between variables.

29. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 29 Objective 5: Example 5.Divorce rates are considerably higher for couples who marry in their teens. The line graphs in the figure show the percentages of marriages ending in divorce based on the wife’s age at marriage.

30. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 30 Objective 5: Example (cont) Here are two mathematical models that approximate the data displayed by the line graphs. Wife is under 18 at time of marriage: Wife is over 25 at time of marriage:

31. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 31 Objective 5: Example (cont) a. Use the appropriate formula to determine the percentage of marriages ending in divorce after 15 years when the wife is under 18 at the time of marriage. 65% of marriages end in divorce after 15 years when the wife is under 18 at the time of marriage.

32. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 32 Objective 5: Example (cont) b. Use the appropriate line graph in the figure to determine the percentage of marriages ending in divorce after 15 years when the wife is under 18 at the time of marriage.

33. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 33 Objective 5: Example (cont) According to the line graph, 60% of marriages end in divorce after 15 years when the wife is under 18 at the time of marriage.

34. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 34 Objective 5: Example (cont) c. Does the value given by the mathematical model underestimate or overestimate the actual percentage of marriages ending in divorce after 15 years as shown by the graph? By how much? The mathematical model overestimates the actual percentage shown in the graph by 5%.