course MTH 158 ¤à·ÌÅ¢ØЕÕݳÊë¦üƱ~‡ÿŽÓŠ—¸assignment #010
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14:28:55 query 1.1.20 (was 1.1.12). Explain, step by step, how you solved the equation 5y + 6 = -18 - y
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RESPONSE --> 5y + 6 = -18 - y 5y + y + 6 = -18 - y + y 6y + 6 = -18 6y + 6 - 6 = -18 - 6 6y = -24 y = -24/6 y = -4 Check: 5(-4) + 6 = -18 - (-4) -20 + 6 = -14 -14 = -14 Because the solution y = -4 satisfies both sides of the equation the solution set is {-4}.
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14:29:55 09-24-2008 14:29:55 ** STUDENT SOLUTION WITH INSTRUCTOR COMMENT: 5y + 6 = 18 - y Subtract 6 from both sides, giving us 5y = 12 - y Add y to both sides, 5y + y = 12 or 6y = 12 divide both sides by 6 y = 2 INSTRUCTOR COMMENT: This is correct for equation 5y + 6 = 18 - y but the equation as I note it is 5y + 6 = -18 - y. If that's the correct equation the solution is found by practically the same steps but you end up with y = -4. **
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NOTES -------> ok
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14:34:01 query 1.1.38 \ 44 (was 1.1.30). Explain, step by step, how you solved the equation (2x+1) / 3 + 16 = 3x
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RESPONSE --> (2x + 1)/3 + 16 = 3x 3[((2x + 1)/3) + 16] = 3(3x) 3((2x + 1)/3) + 3 * 16 = 9x 2x + 1 + 48 = 9x 2x + 49 = 9x 2x - 9x = -49 -7x = -49 x = -49/-7 x = 7 Check: (2(7) + 1)/3 + 16 = 3(7) (14 + 1)/3 + 16 = 21 15/3 + 16 = 21 5 + 16 = 21 21 = 21 Because the solution x = 7 satisfies both sides of the equation the solution set is {7}.
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14:34:28 09-24-2008 14:34:28 ** STUDENT SOLUTION: (2x + 1) / 3 + 16 = 3x First, multiply both sides of the equation by 3 2x +1 + 48 =9x or 2x + 49 = 9x subtract 2x from both sides. 49 = 7x Divide both sides by 7 7 = x **
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NOTES -------> ok
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14:36:37 query was 1.1.44 \ 36. Explain, step by step, how you solved the equation (x+2)(x-3) = (x+3)^2
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RESPONSE --> (x + 2)(x - 3) = (x + 3)^2 x^2 - x - 6 = x^2 + 6x + 9 -x - 6 = 6x + 9 -7x - 6 = 9 -7x = 15 x = -15/7 Check: ((-15/) + 2)((-15/7) - 3) = ((-15/7) + 3)^2 (-1/7)(-36/7) = (6/7)^2 36/49 = 36/49 Because the solution x = -15/7 satisfies both sides of the equation, the solution set is {-15/7}
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14:36:47 09-24-2008 14:36:47 ** STUDENT SOLUTION: (x+2)(x+3) = (x+3)^2 First, we use the distributive property to remove the parenthesis and get x^2 - x - 6 = x^2 + 6x + 9 subtract x^2 from both sides, -x - 6 = 6x + 9 Subtract 9 from both sides - x - 6 - 9 = 6x or -x - 15 = 6x add x to both sides -15 = 7x Divide both sides by 7 -15/7 = x **
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NOTES -------> ok
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14:40:25 query 1.1.52 (was 1.1.48). Explain, step by step, how you solved the equation x / (x^2-9) + 4 / (x+3) = 3 / (x^2-9)/
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RESPONSE --> (x/(x^2 - 9)) +(4/(x + 3)) = (3/(x^2 - 9)) (x/((x + 3)(x - 3))) + (4/(x + 3)) = (3/((x + 3)(x - 3))) ((x + 3)(x - 3)) * [(x/((x + 3)(x - 3))) + (4/(x + 3))] = [(3/((x + 3)(x - 3)))] * ((x + 3)(x - 3)) ((x + 3)(x - 3))(x/((x + 3)(x - 3))) + ((x + 3)(x - 3))(4/(x + 3)) = 3 x + 4(x + 3) = 3 x + 4x - 12 = 3 5x - 12 = 3 5x = 3 + 12 5x = 15 x = 3 Check: (x/(x^2 - 9)) + (4/(x + 3)) = (3/(3^2 - 9)) + (4/(3 + 3)) = 3/0 Because x = 3 causes division by 0, which is undefined, this equation has no solution.
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14:40:59 09-24-2008 14:40:59 ** Starting with x / (x^2 -9) + 4 / (x+3) = 3 / (x^2 -9), first factor x^2 - 9 to get x / ( (x-3)(x+3) ) + 4 / (x+3) = 3 / ( (x-3)(x+3) ). Multiply both sides by the common denominator ( (x-3)(x+3) ): ( (x-3)(x+3) ) * x / ( (x-3)(x+3) ) + ( (x-3)(x+3) ) * 4 / (x+3) = ( (x-3)(x+3) ) * 3 / ( (x-3)(x+3) ). Simplify: x + 4(x-3) = 3. Simplify x + 4x - 12 = 3 5x = 15 x = 3. If there is a solution to the original equation it is x = 3. However x = 3 results in denominator 0 when substituted into the original equation, and division by 0 is undefined. So there is no solution to the equation. When you multiplied both sides by x-3, if x = 3 you were multiplying by zero, which invalidated your solution **
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NOTES -------> ok
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14:45:55 query 1.1.58 (was 1.1.54). Explain, step by step, how you solved the equation (8w + 5) / (10w - 7) = (4w - 3) / (5w + 7)
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RESPONSE --> (8w + 5)/(10w - 7) = (4w - 3)/(5w + 7) ((10w - 7)(5w + 7))[(8w + 5)/(10w - 7)] = [(4w - 3)/(5w + 7)]((10w - 7)(5w + 7)) (5w + 7)(8w + 5) = (4w - 3)(10w - 7) 40w^2 + 25w + 56w + 35 = 40w^2 - 28w - 30w + 21 40w^2 + 81w + 35 = 40w^2 - 58w + 21 81w + 35 = -58w + 21 139w = -14 Check: (8(-14/139) + 5)/(10(-14/139) - 7) = (4(-14/139) - 3)/(5(-14/139) + 7) ((-112/139) + 5)/((-140/139) - 7) = ((-56/139) - 3)/((-70/139) + 7) (583/139)/(-1113/139) = (-473/139)/(903/139) -11/21 = -11/21 Because the solution satisfies both sides of the equation, the solution set is {-14/139}.
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14:46:43 09-24-2008 14:46:43 ** STUDENT SOLUTION: 1) clear the equation of fractions by multiplying both sides by the LCM (10w - 7)(5W + 7) After cancellation the left side reads: (5w+7)(8w + 5) After cancellation the right side reads: (10w - 7)(4w - 3) multiply the factors on each side using the DISTRIBUTIVE LAW Left side becomes: (40w^2) + 81w + 35 Right side becomes: (40w^2) - 58w + 21 3) subtract 40w^2 from both sides add 58w to both sides subtract 35 from both sides Rewrite: 139w = - 14 Now divide both sides by 139 to get w = - (14 / 139) DER**
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NOTES -------> ok
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14:48:13 query 1.1.70 (was 1.1.78). Explain, step by step, how you solved the equation 1 - a x = b, a <> 0.
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RESPONSE --> 1 - ax = b -ax = b - 1 x = (b - 1)/-a x = -(b - 1)/a
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14:49:35 09-24-2008 14:49:35 ** Start with 1 -ax = b, a <> 0. Adding -1 to both sides we get 1 - ax - 1 = b - 1, which we simplify to get -ax = b - 1. Divide both sides by -a, which gives you x = (b - 1) / (-a). Multiply the right-hand side by -1 / -1 to get x = (-b + 1) / a or x = (1 - b) / a. **
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NOTES -------> I never thought to factor the negative into the numerator, I instead was using the negative outside the entire fraction. I do see why it would be easier to understand as 1 + b.
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14:50:12 09-24-2008 14:50:12 ** Start with 1 -ax = b, a <> 0. Adding -1 to both sides we get 1 - ax - 1 = b - 1, which we simplify to get -ax = b - 1. Divide both sides by -a, which gives you x = (b - 1) / (-a). Multiply the right-hand side by -1 / -1 to get x = (-b + 1) / a or x = (1 - b) / a. **
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NOTES -------> I meant (1 - b)/a
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14:57:10 query extra problem (was 1.1.72). Explain, step by step, how you solved the equation x^3 + 6 x^2 - 7 x = 0 using factoring.
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RESPONSE --> This was only briefly mentioned in the text so I'm going basically on memory, which is okay. x^3 + 6x^2 - 7x = 0 x(x^2 + 6x - 7) = 0 x(x + 7)(x - 1) = 0 x = 0 x + 7 = 0 x = -7 x - 1 = 0 x = 1 The solution set would then be, I believe (this is the part I'm foggy on), {-7, 0, 1}.
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14:57:25 09-24-2008 14:57:25 ** Starting with x^3 + 6 x^2 - 7 x = 0 factor x out of the left-hand side: x(x^2 + 6x - 7) = 0. Factor the trinomial: x ( x+7) ( x - 1) = 0. Then x = 0 or x + 7 = 0 or x - 1 = 0 so x = 0 or x = -7 or x = 1. **
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NOTES -------> okay
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15:07:21 query 1.1.90 (was 1.2.18). Explain how you set up and solved an equation for the problem. Include your equation and the reasoning you used to develop the equation. Problem (note that this statement is for instructor reference; the full statement was in your text) scores 86, 80, 84, 90, scores to ave B (80) and A (90).
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RESPONSE --> We are being asked to determine what grade Mike will need to obtain the final grade of 80 (B) or 90 (A). Let f = final exam. His current average is 85. 80 = 85/3 + 2x/3: This is because the current average is counted as 1/3 of his final grade and his exam is 2/3 of his final grade. 80 = 85/3 + (2x)/3 3(80) = [(85/3) + (2x/3)] * 3 240 = 85 + 2x 155 = 2x x = 77.5 90 = 85/3 + 2x + 3 3(90) = [(85/3) + (2x/3)] * 3 270 = 85 + 2x 185 = 2x x = 92.5 In order to obtain a final grade of 80, or a B, Mike must earn an exam grade of 77.5. In order to obtain a final grade of 90, or an A, Mike must earn an exam grade of 92.5.
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15:10:24 09-24-2008 15:10:24 ** This can be solved by trial and error but the only acceptable method for this course, in which we are learning to solve problems by means of equations, is by an equation. Let x be the score you make on the exam. The average of the four tests is easy to find: 4-test average = ( 86 + 80 + 84 + 90 ) / 4 = 340 / 4 = 85. The final grade can be thought of as being made up of 3 parts, 1 part being the test average and 2 parts being the exam grade. We would therefore have final average = (1 * test average + 2 * exam grade) / 3. This gives us the equation final ave = (85 + 2 * x) / 3. If the ave score is to be 80 then we solve (85 + 2 * x) / 3 = 80. Multiplying both sides by 3 we get 85 + 2x = 240. Subtracting 85 from both sides we have 2 x = 240 - 85 = 155 so that x = 155 / 2 = 77.5. We can solve (340 + x) / 5 = 90 in a similar manner. We obtain x = 92.5. Alternative solution: If we add 1/3 of the test average to 2/3 of the final exam grade we get the final average. So (using the fact that the test ave is 85%, as calculated above) our equation would be 1/3 * 85 + 2/3 * x = final ave. For final ave = 80 we get 1/3 * 85 + 2/3 * x = 80. Multiplying both sides by 3 we have 85 + 2 * x = 240. The rest of the solution goes as before and we end up with x = 77.5. Solving 1/3 * 85 + 2/3 * x = 90 we get x = 92.5, as before. **
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NOTES -------> I never thought to combine the current average and the final exam grade (x) all over 3. It should have been obvious. I think I was trying to keep everything seperate, for my mind's sake : )
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15:13:21 query 1.1.82 (was 1.1.90). Explain, step by step, how you solved the equation v = -g t + v0 for t.
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RESPONSE --> v = -gt + v0 v - v0 = -gt (v - v0)/-g = t (-v + v0)/g = t (v0 - v)/g = t So, t = (v0 - v)/g
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15:14:57 09-24-2008 15:14:57 ** NOTE: v0 stands for v with subscript 0; the whole expression v0 stands for the name of a variable. It doesn't mean v * 0. Starting with v = -g t + v0, add -v0 to both sides to get v - v0 = -gt. Divide both sides by -g to get (v - v0) / (-g) = t }or so that t = -(v - v0) / g = (-v + v0) / g.
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NOTES -------> ok
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15:15:05 Query Add comments on any surprises or insights you experienced as a result of this assignment.
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RESPONSE --> k
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15:15:14 **** Query Add comments on any surprises or insights you experienced as a result of this assignment.
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RESPONSE --> k
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