course Mth 163 If your solution to stated problem does not match the given solution, you should self-critique per instructions at
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Given Solution: The system 3a + 3b = 9 6a + 5b = 16 can be solved by adding an appropriate multiple of one equation in order to eliminate one of the variables. Since the coefficient of a in the second equation (the coefficient of a in the second equation is 6)) is double that in the first (the coefficient of a in the first equation is 3), we can multiply the first equation by -2 in order to make the coefficients of a equal and opposite: -2 * [ 3a + 3b ] = -2 [ 9 ] 6a + 5b = 16 gives us -6a - 6 b = -18 6a + 5b = 16 . Adding the two equations together we obtain -b = -2, or just b = 2. Substituting b = 2 into the first equation we obtain 3 a + 3(2) = 9, or 3 a + 6 = 9 so that 3 a = 3 and a = 1. Our solution is therefore a = 1, b = 2. We used the first equation in our last step, so we verigy this solution is by substituting these values into the second equation, where we get 6 * 1 + 5 * 2 = 6 + 10 = 16. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Self-critique Rating: 2 ********************************************* Question: `q002. Solve the following system of simultaneous linear equations using the method of elimination: 4a + 5b = 18 6a + 9b = 30. ********************************************* Your solution: 4a + 5b = 18 6a + 9b = 30 First we have to see what they have in common. 4 and 6 both go into 12 so we have to multiply them to get 12. 3 (4a + 5b ) = 3 (18) -2 (6a +9b) = -2 (30) Then 12 a + 15 b = 54 -12 a – 18 b = 60 Add them together -3b = -6 B= 2 Confidence Assessment:
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Given Solution: In the system 4a + 5b = 18 6a + 9b = 30 we see that the coefficients of b are relatively prime; they therefore have a least common multiple equal to 5 * 9. The coefficients 4 and 6 of a have a least common multiple of 12. • We have a choice of which variable to eliminate. We could 'match' the b by multiplying the first equation by 9 and the second by -5, or we could match the coefficients of a by multiplying the first equation by 3 and the second by -2. • Either choice would work. The numbers required to 'match' the coefficients of a are smaller, but the numbers required to 'match the coefficients of b would otherwise work equally well. Choosing to 'match' the coefficient of a, we obtain 3 * [4a + 5b ] = 3 * 18 -2 * [ 6a + 9b ] = -2 * 30, so the system becomes 12 a + 15 b = 54 -12 a - 18 b = -60. Adding the equations we get -3 b = -6, so b = 2. Substituting this value of b into the first equation we obtain 4 a + 5 * 2 = 18, or 4 a + 10 = 18, which we easily solve to obtain a = 2. Substituting this value of a into the second equation we obtain 6 * 2 + 9 * 2 = 30, which verifies our solution. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Self-critique Rating: ********************************************* Question: `q003. If y = 5x + 8, then for what value of x will we have y = 13? ********************************************* Your solution: To find that y = 13 you have to plug in y So the equation would read 13= 5x + 8 Then subtract 8 on both sides to get x alone. Then divide by x 13 – 8 = 5 5 = 5x 5/5x = 1 X= 1 Confidence Assessment:
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Given Solution: We first substitute y = 13 into the equation y = 5 x + 8 to obtain 13 = 5 x + 8. Subtracting 8 from both equations and reversing the equality we obtain 5 x = 5, which we easily solve to obtain x = 1. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Self-critique Rating: ********************************************* Question: `q004. Sketch a set of coordinate axes representing y vs. x, with y on the vertical axis and x on the horizontal axis. • Plot the points (1, -2), (3, 5) and (7, 8). • Sketch a smooth curve passing through these three points. On your curve, what are the y coordinates corresponding to x coordinates 1, 3, 5 and 7? Estimate these coordinates as accurately as you can from your graph. Retain your sketch for use in future questions. ********************************************* Your solution: With the x coordinates 1,3, and 7 the y coordinates are -2, 3, and 8 Confidence Assessment:
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Given Solution: The x coordinates 1, 3 and 7 match the x coordinates of the three given points, the y coordinates will be the y coordinates -2, 3 and 8, respectively, of those points. • At x = 5 the precise value of x, for a perfect parabola, would be 8 1/3, or about 8.333. • You are unlikely to have drawn a perfect parabola and your estimate will almost certainly differ from this value. Any estimate with .5 or so of this value would be a good estimate. Drawn with complete accuracy a parabola through these points will peak between x = 3 in and x = 7. • The peak of the actual parabola will occur close to x = 6. Any estimate between x = 5 and x = 7 is pretty good, and even an estimate between x = 7 and x = 8 is not unreasonable. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Self-critique Rating: ********************************************* Question: `q005. Using your sketch from the preceding exercise, estimate the x coordinates corresponding to y coordinates 1, 3, 5 and 7. Also estimate the x values at which y is 0. ********************************************* Your solution: Y= 1, 3, 5, 7 X= 1, 2 ,3 ,4 When y is 0 x = 9, 5, 7 Confidence Assessment:
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Given Solution: The easiest way to estimate your points would be to make horizontal lines on your graph at y = 1, 3, 5 and 7. You would easily locate the points were these lines intersect your graph, then estimate the x coordinates of these points. For the actual parabola passing through the given points, y will be 1 when x = 1.7 (and also, if your graph extended that far, near x = 10). • y = 3 near x = 2.3 (and near x = 9.3). • y = 5 at the given point (3, 5), where x = 3. • y = 7 near x = 4 (and also near x = 7.7). Any set of estimates in the vicinity of these values is good. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Self-critique Rating: ********************************************* Question: `q006. Suppose the graph you used in the preceding two exercises represents the profit y on an item, with profit given in cents, when the selling price is x, with selling price in dollars. According to your graph: • What would be the profit if the item is sold for 4 dollars? • What selling price would result in a profit of 7 cents? • Why is this graph not a realistic model of profit vs. selling price? ********************************************* Your solution: X= 4 when put on the graph followed up the graph it is closest to 7 If the profit was 7 cents then the selling price would be 7.7 This isn’t a realistic model because it doesn’t show how profit and selling prices go together, it doesn’t match up. Confidence Assessment:
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Given Solution: To find the profit for a selling price of x = 4 dollars, we would look at the x = 4 point on the graph. • This point is easily located by sketching a vertical line through x = 4. Projecting over to the y-axis from this point, you should have obtained an x value somewhere around 7, representing 7 cents. The profit is the y value, so to obtain the selling price x corresponding to a profit of y = 7 we sketch the horizontal line at y = 7, which as in a preceding exercise will give us x values of about 4 (y = 7 would also occur at x = 7.7, approx., if the entire parabola was drawn). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Self-critique Rating: ********************************************* Question: `q007. On another set of coordinate axes, plot the points (-3, 4) and (5, -2). Sketch a straight line through these points. We will proceed step-by-step obtain an approximate equation for this line: First substitute the x and y coordinates of the first point into the form y = m x + b. • What equation do you obtain when you make this substitution? ********************************************* Your solution: ( -3 , 4) x= -3 y = 4 4= -3m +b Confidence Assessment:
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Given Solution: Substituting x = -3 and y = 4 into the form y = m x + b, we obtain the equation • 4 = -3 m + b. We can reverse the right- and left-hand sides to get • -3 m + b = 4. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Self-critique Rating: ********************************************* Question: `q008. Substitute the coordinates of the point (5, -2) into the form y = m x + b. What equation do you get? ********************************************* Your solution: X= 5 y= -2 -2= m 5 + b Confidence Assessment:
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Given Solution: Substituting x = 5 and y = -2 into the form y = m x + b, we obtain the equation • -2 = 5 m + b. Reversing the sides we have • 5 m + b = -2 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Self-critique Rating: ********************************************* Question: `q009. You have obtained the equations -3 m + b = 4 and 5 m + b = -2. Use the method of elimination to solve these simultaneous equations for m and b. ********************************************* Your solution: Just subtract the first equation from the second -3 m + b = 4 - 5m + b = -2 -8 m = 6 M= -3/4 Substitute b (-3/4) -3 + b= 4 B= 7/4 Confidence Assessment:
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Given Solution: Starting with the system -3 m + b = 4 5 m + b = -2 we can easily eliminate b by subtracting the equations. If we subtract the first equation from the second we obtain -8 m = 6, with solution m = -3/4. Substituting this value into the first equation we obtain (-3/4) * -3 + b = 4, which we easily solve to obtain b = 7/4. To check our solution we substitute m = -3/4 and b = 7/4 into the second equation, obtaining 5 ( -3/4) + 7/4 = -2, which gives us -8/4 = -2 or -2 = -2. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Self-critique Rating: ********************************************* Question: `q010. Substitute your solutions b = 7/4 and m = -3/4 into the original form y = m x + b. • What equation do you obtain? • What is the significance of this equation? ********************************************* Your solution: Y= -3/4 + 7/4 Equation of a straight line through points (-3, 4) and (5, -2) Confidence Assessment:
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Given Solution: Substituting b = 7/4 and m = -3/4 into the form y = m x + b, we obtain the equation y = -3/4 x + 7/4. This is the equation of the straight line through the given points (-3, 4) and (5, -2).