1. Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 2.3 - 1 2.3 Applications of Linear Equations

2. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.3 - 2 2.3 Applications of Linear Equations Problem-Solving Hint PROBLEM-SOLVING HINT Usually there are key words and phrases in a verbal problem that translate into mathematical expressions involving addition, subtraction, multiplication, and division. Translations of some commonly used expressions follow.

3. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.3 - 3 2.3 Applications of Linear Equations Translating from Words to Mathematical Expressions Verbal Expression The sum of a number and 2 Mathematical Expression (where x and y are numbers) Addition 3 more than a number 7 plus a number 16 added to a number A number increased by 9 The sum of two numbers x + 2 x + 3 7 + x x + 16 x + 9 x + y

4. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.3 - 4 2.3 Applications of Linear Equations Translating from Words to Mathematical Expressions Verbal Expression 4 less than a number Mathematical Expression (where x and y are numbers) Subtraction 10 minus a number A number decreased by 5 A number subtracted from 12 The difference between two numbers x – 4 10 – x x – 5 12 – x x – y

5. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.3 - 5 2.3 Applications of Linear Equations Translating from Words to Mathematical Expressions Verbal Expression 14 times a number Mathematical Expression (where x and y are numbers) Multiplication A number multiplied by 8 Triple (three times) a number The product of two numbers 14x 8x8x 3x3x xy of a number (used with fractions and percent) 3 4 x 3 4

6. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.3 - 6 2.3 Applications of Linear Equations Translating from Words to Mathematical Expressions Verbal Expression The quotient of 6 and a number Mathematical Expression (where x and y are numbers) Division A number divided by 15 The ratio of two numbers or the quotient of two numbers (x ≠ 0) 6 x (y ≠ 0) x y x 15

7. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.3 - 7 CAUTION Because subtraction and division are not commutative operations, be careful to correctly translate expressions involving them. For example, “5 less than a number” is translated as x – 5, not 5 – x. “A number subtracted from 12” is expressed as 12 – x, not x – 12. For division, the number by which we are dividing is the denominator, and the number into which we are dividing is the numerator. For example, “a number divided by 15” and “15 divided into x ” both translate as. Similarly, “the quotient of x and y ” is translated as. 2.3 Applications of Linear Equations Caution x 15 x y

8. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.3 - 8 2.3 Applications of Linear Equations Indicator Words for Equality Equality The symbol for equality, =, is often indicated by the word is. In fact, any words that indicate the idea of “sameness” translate to =.

9. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.3 - 9 2.3 Applications of Linear Equations Translating Words into Equations Verbal SentenceEquation If the product of a number and 16 is decreased by 25, the result is 87. The quotient of a number and the number plus 6 is 48. The quotient of a number and 8, plus the number, is 54. Twice a number, decreased by 4, is 32.

10. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.3 - 10 2.3 Applications of Linear Equations Distinguishing between Expressions and Equations (a) 4(6 – x) + 2x – 1 (b) 4(6 – x) + 2x – 1 = –15 There is no equals sign, so this is an expression. Because of the equals sign, this is an equation. Decide whether each is an expression or an equation.

11. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.3 - 11 Solving an Applied Problem Step 1 Read the problem, several times if necessary, until you understand what is given and what is to be found. Step 2 Assign a variable to represent the unknown value, using diagrams or tables as needed. Write down what the variable represents. Express any other unknown values in terms of the variable. Step 3 Write an equation using the variable expression(s). Step 4 Solve the equation. Step 5 State the answer to the problem. Does it seem reasonable? Step 6 Check the answer in the words of the original problem. 2.3 Applications of Linear Equations Six Steps to Solving Application Problems

12. 2.3 Applications of Linear Equations Solving a Geometry Problem The length of a rectangle is 2 ft more than three times the width. The perimeter of the rectangle is 124 ft. Find the length and the width of the rectangle.

13. 2.3 Applications of Linear Equations Finding Unknown Numerical Quantities A local grocery store baked a combined total of 912 chocolate chip cookies and sugar cookies. If they baked 336 more chocolate chip cookies than sugar cookies, how many of each did the store bake?

14. 2.3 Applications of Linear Equations Solving a Percent Problem During a 2-day fundraiser, a local school sold 1440 raffle tickets. If they sold 350% more raffle tickets on the second day than the first day, how many raffle tickets did they sell on the first day?

15. 2.3 Applications of Linear Equations Solving an Investment Problem A local company has $50,000 to invest. It will put part of the money in an account paying 3% interest and the remainder into stocks paying 5%. If the total annual income from these investments will be $2180, how much will be invested in each account?

16. 2.3 Applications of Linear Equations Solving a Mixture Problem A chemist must mix 12 L of a 30% acid solution with some 80% solution to get a 60% solution. How much of the 80% solution should be used?

17. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.3 - 17 2.3 Applications of Linear Equations Problem-Solving Hint PROBLEM-SOLVING HINT When pure water is added to a solution, remember that water is 0% of the chemical (acid, alcohol, etc.). Similarly, pure chemical is 100% chemical.

18. 2.3 Applications of Linear Equations Solving a Mixture Problem A chemist must mix 8 L of a 10% alcohol solution with pure alcohol to get a a 40% solution. How much of the pure alcohol solution should be used?