Evaluating Expressions

Transcription

1 -1

2 0

3 Chapter 8 Evaluating Expressions c H. Feiner Definition of a Variable Expression A variable expression is a collection of numbers, letters (variables), operations, grouping symbols, any mathematical symbols except an equal sign. Pieces of an expression related by + or are terms. Pieces of an expression related through a multiplication sign are factors. Examples of expressions: a + 5 3x 4y x 3 y 3 x y V πx y (a + b) a b A h (B + b) A rational expression is a ratio (fraction) of two expressions. The term is usually reserved for the ratio of two polynomials (to be defined in another chapter.) x + 6xy + 5y x + y 1

4 CHAPTER 8. EVALUATING EXPRESSIONS 8. Evaluation of Expressions To evaluate means to find the value of something. Evaluating a if a = 5 means finding the value of twice the number in the a box which is known (in this example to be 5. OPen a set of parentheses in place of the variable a, then drop the value of a = into these parentheses. Thus a = (5) = 10. Example 1: Evaluate a + 5 if a = 3. a + 5 = () + 5 = ( 3) + 5 = = 1 Example : Evaluate 3x 4(y 3) if x = and y = 3. 3x 4(y 3) = 3() 4[() 3] = 3( ) 4[( 3) 3] = 3(4) 4( 3 3) = 1 4( 6) = = 36 Example 3: Evaluate x3 y 3 x y if x = 4 and y =. x 3 y 3 x y = ()3 () 3 () () = (4)3 () 3 (4) () = = 56 0 = undefined Example 4: Evaluate V πx y if V = 3, 140, x = 10 and y = 3. Approximate π = 3.14.

5 8.. EVALUATION OF EXPRESSIONS 3 V πx y = () ()() () = (3, 140) (3.14)(10) (3) = 3, 140 (3.14)(100)(3) = 3, 140 (314)(3) = 3, =, 198 Example 5: Evaluate (a + b) a b if a = 6 and b = 6. (a + b) a b = [() + ()] () () = [(6) + ( 6)] (6) ( 6) = (0) = 7 Example 6: Evaluate A h (B + b) if A = 100, h = 40, B = 1 and b = 8. A h (B + b) () = () [() + ()] (40) = (100) [(1) + (8)] = (0) = 100 (1) = = 99 Example 7: Evaluate x + 6xy + 5y x + y if x = 0.4 and y = 1.5.

6 4 CHAPTER 8. EVALUATING EXPRESSIONS x + 6xy + 5y x + y = () + 6()() + 5() () + () = (0.4) + 6(0.4)(1.5) + 5(1.5) (0.4) + (1.5) (.4)(1.5) + 5(.5) = = = = 3.4 = = 5 Example 8: Evaluate 4x 4 x if x = 1. 4x 4 x = 4() 4 () ( ) 4 1 = 4 ( 1 = 4 16 = ( ) ) = 7 9 Example 9: Evaluate y y y 1 x x 1 (x x 1 ) if x = 4, x 1 =, x = 5, y = 10 y 1 = 9 and y = 7.

7 8.. EVALUATION OF EXPRESSIONS 5 y y y 1 () () (x x 1 ) = () [() ()] x x 1 () () (7) (9) = (10) [(4) (10)] ( 5) ( ) = 10 3 ( 6) Example 10: = 10 1 () = 10 ( )() = = 14 A logging company cut a certain number (say T) of trees on Monday. On Tuesday the company cut 5 more trees than on Monday. On Wednesday the number of trees harvested was twice the number of Tuesday. On Thursday the number was half of the number on Monday. (a) Write an expression for the total number of trees cut on the four days. (b) If trees were cut down on Monday, what was the total number of trees harvested on the four days? Trees cut on Monday T Trees cut on Tuesday T + 5 Trees cut on Wednesday (T + 5) Trees cut on Thursday T (a) Total number of trees cut:t + (T + 5) + (T + 5) + T if T =. (b) Evaluate + ( + 5) + ( + 5) + = () + (5) + = + ( + 5) + ( + 5) + = () + (5) + = = = = = = 114 trees. Example 11: Abel works 14 hours in a particular week. Bianca works 1 hours in that week. Abel gets paid $a per hour and Bianca earns $b per hour.

8 6 CHAPTER 8. EVALUATING EXPRESSIONS (a) Write an expression for the total wages Abel and Bianca earn in that week. (b) Then evaluate that expression if Abel gets $8 per hour and Bianca earns $1 per hour. (a) Abel and Bianca earn 14a + 1b dollars in that week. (b) Evaluating 14a + 1b leads to 14() + 1() = 14(8) + 1(1) = = $ Like Terms Occasionally terms look alike in an expression. These like-terms can and should be combined. + 3 = 5. apples added to three apples results in 5 apples. x + 3x = (x + x) + (x + x + x) = 5x x + 3x = (x + x ) + (x + x + x ) = 5x Don t confuse with (x )(3x ) = ()(3)(x x)(x x) = 6x 4 Example 1: Combine like terms: x 3 + 5x x 3 + x 9 x 3 + 5x x 3 + x 9 = x 3 + 5x x 3 + x 9 mark each like-term with its own symbol = (x 3 + 6x 3 ) + (5x + x) + (9 9) = 11x 3 + 6x + 0 Example 13: Combine like terms: 7x 3 + 4[5(x + 8) + 6x 3 x ]

9 8.4. EXERCISES 8 7 7x 3 + 4[5(x + 8) + 6x 3 x ] = 7x 3 + 4[5(x + 8) + 6x 3 x ] Remember PEMDAS? Focus on the innermost group (parentheses). Addition is the only operation. We cannot add an unknown (we do not know the value) to a constant (a known non-varying number). = 7x 3 + 4[5x + 5(8) + 6x 3 x ] Use the distributive property of multiplication over addition to = 7x 3 + 4[5x x 3 x ] Underline like-terms. = 7x 3 +4[(5x x) + (40 ) + 6x 3 ] Associate like-terms. remove the parentheses. Remember that x is a number. = 7x 3 + 4[4x x 3 ] Compute. = 7x 3 + 4[6x 3 + 4x + 38] Rewrite with exponents in decreasing order. = 7x 3 + 4(6x 3 ) + 4(4x) + 4(38) Distribute multiplication over addition. = 7x 3 + 4x x + 15 Compute. 8.4 Exercises 8 1. Evaluate P (L + W) if P = 50, L = 9 and W = 4.. Evaluate S (x + xy) if S = 100, x = 3 and y = Evaluate x b + b 4ac a if x = 10, a = 1, b = 4 and c = Evaluate D 16t + vt + h if D = 00, t = 3, v = 0 and Evaluate S a + b if S = 169, a = 1 and b = Evaluate y y y 1 x x 1 (x x 1 ) if x = 1, x 1 = 9, x = 6, y = 19 y 1 = 7 and y = A bus travels a certain number (say x) of miles on Interstate 405. Then on highway 5 the bus travels 17 more miles than on Interstate 405. The bus now turns to highway 18 and covers three times as many miles than on highway 18. The trip ends on local streets with the number of miles one quarter the number on Interstate 405. (a) Write an expression for the total number of miles traveled. (b) If 3 miles were traveled on Interstate 405, what was the total mileage? 8. Candy studies 144 pages for a test. Diane studies 10 pages for the same test. Candy gets c problems done per page studied and Diane finishes d problems per page. (a) Write an expression for the total number of problems Candy and Diane solve for the test. (b) Then evaluate that expression if Candy completes 3 problems per page and Diane succeeds in finishing 4 problems per page.

10 8 CHAPTER 8. EVALUATING EXPRESSIONS 9. Simplify 7x x 4x x 4 11x 3 + 4x Combine like terms: 9x 4 + 7[3(x + 1) 6x 4 x + ] STOP! 1. Evaluate P (L + W) if P = 50, L = 9 and W = 4. P (L + W) = () [() + ()] = (50) [(9) + (4)] = 50 (18 + 8) = 50 6 = 4. Evaluate S (x + xy) if S = 100, x = 3 and y = 5. S (x + xy) = () [() + ()()] = (100) [( 3) + ( 3)(5)] = 100 [(9) + ( 6)(5)] = 100 [18 + ( 30)] = 100 ( 1) = Evaluate x b + b 4ac a if x = 10, a = 1, b = 4 and c = 1. x b + b 4ac a = () () + () 4()() () = (10) ( 4) + ( 4) 4(1)( 1) (1) = ( 1) = = = = = 10 7 = 3

11 8.4. EXERCISES Evaluate D 16t + vt + h if D = 00, t = 3, v = 0 and 18. D 16t + vt + h = () 16() + ()() + () = (00) 16(3) + (0)(3) + (18) = 00 16(9) = = = = Evaluate S a + b if S = 169, a = 1 and b = 5. S a + b = () () + () = (169) (1) + ( 5) = = = Evaluate y y y 1 (x x 1 ) if x = 1, x 1 = 9, x = 6, y = 19 y 1 = 7 and y = 10. x x 1 y y y 1 () () (x x 1 ) = () x x 1 () () [(x) (x 1)] = (19) ( 10) ( 7) [(1) (9)] (6) (9) = (1 9) 6 9 = (3) = 19 (1)(3) = A bus travels a certain number (say x) of miles on Interstate 405. Then on highway 5 the bus travels 17 more miles than on Interstate 405. The bus now turns to highway 18 and covers three times as many miles than on Interstate 405. The trip ends on local streets with the number of miles one quarter the number on Interstate 405. (a) Write an expression for the total number of miles traveled. (b) If 3 miles were traveled on Interstate 405, what was the total mileage? Miles traveled on Interstate 405 x Miles traveled on highway 5 x + 17 Miles traveled on highway 18 3(x + 17) x Miles covered on local streets 4 (a) Total number of miles: x + (x + 17) + 3(x + 17) + x 4. (b) Evaluate if x = 3

12 10 CHAPTER 8. EVALUATING EXPRESSIONS x + (x + 17) + 3(x + 17) + x 4 = () + [(x) + 17] + 3[() + 17] + () 4 = (3) + [(3) + 17] + 3[(3) + 17] + (3) 4 = (49) + 8 = = = Candy studies 144 pages for a test. Diane studies 10 pages for the same test. Candy gets c problems done per page studied and Diane finishes d problems per page. (a) Write an expression for the total number of problems Candy and Diane solve for the test. (b) Then evaluate that expression if Candy completes 3 problems per page and Diane succeeds in finishing 4 problems per page. (a) Candy and Diane complete 144c + 10d problems for the test. (b) Evaluating 144c + 10d leads to 144c + 10d = 144() + 10() = 144(3) + 10(4) = = Simplify 7x x 4x x 4 11x 3 + 4x + 9 7x x 4x x 4 11x 3 + 4x + 9 = 7x x }{{} 4x + }{{} 9 +3x 4 11x 3 + }{{} 4x + }{{} 9 }{{}}{{} = 7x 4 + 3x 4 11x x }{{} 4x + }{{} 4x + }{{} 9 + }{{} 9 }{{}}{{} = (7x 4 + 3x 4 ) 11x x + ( 4x + 4x) + (9 + 9) = 10x 4 11x x Combine like terms: 9x 4 + 7[3(x + 1) 6x 4 x + ]

13 8.4. EXERCISES x 4 + 7[3(x + 1) 6x 4 x + ] = 9x 4 + 7[3x + 3 6x 4 x + ] = 9x 4 4x x + 35 = 9x 4 + 7[3x + 3 6x 4 x + ] = 9x 4 4x x + 35 = 9x 4 + 7[ 6x 4 + 3x x ] = 9x 4 + 7[ 6x 4 + (3x x) + (3 + )] = 9x 4 + 7[ 6x 4 + x + 5] = 9x 4 + 7( 6x 4 ) + 7(x) + 7(5) = (9x 4 4x 4 ) + 14x + 35 = 33x x + 35