CH 9 INTRO TO EQUATIONS

Transcription

1 CH 9 INTRO TO EQUATIONS INTRODUCTION I m thinking of number. If I dd 10 to the number, the result is 5. Wht number ws I thinking of? R emember this question from Chpter 1? Now we re redy to formlize the thought processes you went through to solve this problem. It my seem silly to mke big del bout this esy problem, but the skills you develop now will give you the tools needed to solve problems tht could never be solved in your hed. An eqution must lwys be in blnce. ONE-STEP EQUATIONS Considering the problem gin, let s pretend we don t lredy know the nswer. We ll choose some symbol to represent the unknown number; cll it n. Here s wht we need to do: Trnslte the English sttement If I dd 10 to the number, the result is 5 into n eqution, sttement of equlity. Since we let n represent the number, then the phrse If I dd 10 to the number becomes n + 10, the phrse the result is becomes =, nd the 5 is, of course, just 5. The finl trnsltion from English to lgebr results in the eqution n 10 = 5 If I dd 10 5 the result to the number is An eqution is like seesw in blnce, perfectly level with child t ech end. If we were to drop bowling bll onto the lp of one child, tht child would plummet to the ground, throwing the second child into

2 2 the ir -- ll in ll, not plesnt sitution. But most importntly, the seesw would not be in blnce nymore -- one end would be on the ground nd the other up in the ir. But wht if we dropped 10-lb bowling bll on the lp of ech child t exctly the sme moment? Even with kids screming in pin, t lest the seesw would remin level, still in blnce. By the wy, mth techers don t cre bout kids in pin, s long s the eqution remins blnced! This is the essence of solving n eqution: As long s we perform the sme opertion on ech side of n eqution, the resulting sttement should still be blnced eqution. Thus, to solve the eqution bove, n + 10 = 5, we need to do something to ech side of the eqution tht will tell us wht n is. We hve to isolte the n. To do tht, we hve to strip wy the 10 from the expression n Here's the thought process: The opertion connecting the n with the 10 is ddition. The opertion which removes ddition is subtrction. Therefore, we need to subtrct 10 from the left side of the eqution, which will leve the n ll by itself (isolted). But whtever we do to one side of n eqution, we must do to the other side. n + 10 = 5 (the originl eqution) n = 5 10 (subtrct 10 from ech side of the eqution) n = 5 (simplify ech side) Therefore, the number I ws thinking of ws 5

3 The key to solving ny eqution: DO THE SAME THING TO EACH SIDE OF THE EQUATION. EXAMPLE 1: Solve ech eqution: A. x + 17 = 50 Since the x nd the 17 re connected by ddition, we will use subtrction to remove the 17. Remembering to lwys do the sme thing to ech side of the eqution, we get x = x = (subtrct 17 from ech side) (simplify ech side) B. y 12 = 1 In this eqution, the 12 is being subtrcted from the unknown y; to undo the subtrcting, we dd 12 to ech side of the eqution: y = y = 25 (dd 12 to ech side) (simplify ech side) C. 7n = 5 This time the opertion to remove is multipliction, which is esily removed by division: 7n (divide ech side by 7) n = 5 (simplify ech side)

4 D. = 2 8 The opertion connecting the unknown with the 8 is division. Since the reverse of division is multipliction, we shll multiply ech side of the eqution by 8: 8 = (2) 8 (multiply ech side by 8) 8 8 = 16 (cross-cncel the 8 s) 8 = 16 (simplify) This is wht hppens if you don t do the sme thing to ech side of the eqution. Homework 1. Solve ech eqution:. w + 17 = 25 b. 9 = c. x 2 = 90 d. y + 99 = 105 e. z 2 = 80 f. b + 50 = 160 g. 8 + x = 12 h t = 100 i. 0 + m = 0

5 5 j. n 17 = 50 k. J + 20 =.9 l.. + T = 100 m. w.5 = 8.2 n. u + = 1000 o. n = Solve ech eqution:. x = 18 b. 2n = 1 c. 5u = 95 d. z = 0. e. 1.m = f. 0.25Q = 0.5 g. 7x = 28 h. 2 = 128 i. 5R = 1000 j. 1000k = 10,000 k. 2x = 17 l. 7h = 1 m. 12c = 12 n. 18 = 0 o. 2x = 10. Solve ech eqution:. x = 6 d. = 5 2 g. x =1 7 y j. = y b. = 2 e. b = 17 h. = 0 0 p k. = c. z = 2 f. c =1 88 i. u = 17 q l. = Solve ech eqution:. 2x = 18 b. y = 16 c. n = d. 7 + x = 9 e. x = 2 f. y + 9. = 9. g. n(7) = 9 h. z 1 = 1 i = 9.1 j. b 0.7 = 2 k. c + =. l. d 2.1 = m. x = 7 y n. =1 9 o. z =

6 6 TWO-STEP EQUATIONS Here s nother word problem tht cn be done by guessing or working bckwrds: I m thinking of number. If I multiply it by nd then dd, the result is. Wht is the number I ws thinking of? Trnslting this problem into lgebr (ssuming n is the unknown number) results in the eqution n + = As we lerned bove, since the is connected to the n by multiplying, we rid ourselves of the by dividing. And becuse the is being dded to the n, we remove the by subtrcting. And, of course, whenever we do something to the left side of the eqution, we ll keep things in blnce by doing the sme thing to the right side. This leves one importnt question: Which do we get rid of first, the or the? Just like unwrpping gift, we strt with the finl opertion in the expression, nd work bckwrds until the vrible n is lone. The finl opertion in the expression n + is ddition, so we get rid of the first: n + = (the originl eqution) n + = (subtrct from ech side) n = 0 (simplify ech side) n = 0 (divide ech side by ) n = 10 (simplify ech side) nd we ve found tht the unknown number is 10

7 7 Note: Since the expression n + ws creted using the Order of Opertions -- multiply, then dd -- we undo the order of opertions by going in the reverse order: First we undo the lst opertion, ddition, nd second we undo the first opertion, multipliction. The next exmple shows us how we generlly leve frctionl nswers in lgebr. EXAMPLE 2: Solve for x: 1 + 8x = 22 Solution: Add 1 to ech side of the eqution: x = Simplify: 8x = 6 Divide ech side by 8: 8x = Simplify: x 9 = 2 Notice tht it s oky in lgebr to leve n improper frction s your finl nswer, s long s it s in reduced form. In this exmple, the nswer 6 8 converted to mixed number. must be reduced to 9, but it need not be 2 Some students find it esier by first using the commuttive property of ddition to flip the terms on the left side of the given eqution: 8x 1 = 22 EXAMPLE : Solve for : 2 = 10 Solution: We notice tht in this eqution, the first opertion on the vrible is the division, followed by the subtrction. Therefore, our pln is to reverse these steps.

8 8 The eqution we re to solve: 2 = 10 Add 2 to ech side of the eqution: = And simplify: = 12 Lst, multiply ech side of the eqution by : = 12 And we re done: = 8 EXAMPLE : Solve for n: n+ 7 = 9 Solution: In this eqution the finl opertion is division, so our first step in solving the eqution is to multiply. n 7 = 9 (multiply ech side by ) n + 7 = 27 (simplify ech side) n = 27 7 (subtrct 7 from ech side) n = 20 (simplify ech side) Homework 5. Solve ech eqution, leving nswers s frctions:. n + 7 = 22 b. 2x 1 = 9 c. q + 17 = 9 d. 7z 7 = 7 e. 1w + 26 = 65 f. 2x + 18 = 8 g. 18u 7 = 11 h. 12v + 80 = 80 i. 7 + = 10

9 9 j. n + 17 = 17 k. 2T = 55 l k = 20 m. 7x + 12 = 100 n. y 1 = 20 o. 7t + 18 = 18 p. 1c 98 = 1 q h = 2 r w = Solve ech eqution:. d. g. j. x 5 = 2 c = 0 n = 5 d 7 = 0 b. = 5 c. u 1 = y e. 1 = 5 f. w 7 = h. b = 1 i. z 1 = k. m g 2 = 5 l. 70 = Solutions 1.. w = 8 b. = 1 c. x = 92 d. y = 6 e. z = 112 f. b = 110 g. x = h. t = 88 i. m = 0 j. n = 67 k. J = 2.9 l. T = 96.7 m. w = 12.7 n. u = 996 o. n = x = 6 b. n = 72 c. u = 19 d. z = 7.6 e. m = 99 f. Q = 1.8 g. x = h. = i. R = 200 j. k = 10 k. x = 8.5 l. h = 2 m. c = 1 n. = 0 o. x = 5.. x = 18 b. y = 96 c. z = 66 d. = 115 e. b = 578 f. c = 88 g. x = 7 h. = 900 i. u = 51 j. y = 10 k. p = 0.68 l. q = 5.29

10 10.. x = 9 b. y = 5. c. n = 0.75 d. x = 2 e. x = 5 f. y = 0 g. n = 7 h. z = 2 i. = 6.8 j. b = 2.7 k. c = 1. l. d = 20.9 m. x = 21 n. y = 9 o. z = For this problem, nswers will be written in frction form, just for vriety. Notice tht it s oky in lgebr to leve n improper frction s your finl nswer, s long s it s been reduced. For exmple, the nswer 10 8 must be reduced to 5, but it need not be converted to mixed number.. n = 5 b. x = 25 c. q = 19 d. z = 2 e. w = f. 20 x = g. u = 1 h. v = 0 2 i. = 1 j. n = 0 k. 99 T = l. 2 m. 88 x = n. y = 11 o. t = 0 p. 7 q. h = 1 r. w = x = 6 b. = 27 c. u = 10 d. c = e. y = 28 f. w = 2 g. n = 12 h. b = 16 i. z = 17 j. d = 7 k. m = 18 l. g = 1,10 If ntion expects to be ignornt nd free, it expects wht never ws, nd will never be. Thoms Jefferson k c = =