course Mth 163
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20:08:45 `q001. Note that this assignment has 10 questions Solve the following system of simultaneous linear equations: 3a + 3b = 9 6a + 5b = 16.
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RESPONSE --> 3a + 3b = 9 6a + 5b = 16 multiply the first one by -2 -6a - 6b =-18 6a + 5b = 16 eliminate the a -1b = -2 b=2 Then plug that in 3a + 6 = 9 3a = 3 a =1
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20:09:03 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. This solution is verified by substituting these values into the second equation, where we get 6 * 1 + 5 * 2 = 6 + 10 = 16.
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RESPONSE --> I got it
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20:17:18 `q002. Solve the following system of simultaneous linear equations using the method of elimination: 4a + 5b = 18 6a + 9b = 30.
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RESPONSE --> take the problem and multiply the first one by -6 and the second one by 4 -24 -30b = -108 24 + 36b = 120 the a's cancel each other abd we are left with 6b = 12 b=2 plug it in and you have 4a + 10 = 18 4a = 8 a = 2 a= 2, b=2
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20:19:38 In the system 4a + 5b = 18 6a + 9b = 30 we see that the coefficients of b are relatively prime and so have a least common multiple equal to 5 * 9, whereas the coefficients 4 and 6 of a have a least common multiple of 12. We could therefore 'match' the coefficients of a and b by multiplying the first equation by 9 in the second by -5 in order to eliminate b, or by multiplying the first equation by 3 and the second by -2 in order to eliminate a. Choosing the latter in order to keep the number smaller, we obtain 3 * [4a + 5b ] = 3 * 18 -2 * [ 6a + 9b ] = -2 * 30, or 12 a + 15 b = 54 -12 a - 18 b = -60. Adding the two we get -3 b = -6, so b = 2. Substituting this value 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.
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RESPONSE --> i got the same answer, but I didn't use the least common multiple.
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20:22:01 `q003. If y = 5x + 8, then for what value of x will we have y = 13?
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RESPONSE --> 13 = 5x + 8 subtract 8 from each side 5= 5x x = 1
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20:22:12 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.
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RESPONSE --> i got it
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20:31:39 `q004. Sketch a set of coordinate axes representing y vs. x, with y on the vertical axis and x on the horizontal axis, and 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 assignments.
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RESPONSE --> I used the same cooridantes given, (1, -2), (3,5), (7,8) but I estimated from my graph (5,7) based on my sketch. Also the line represents decreasing
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20:37:00 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. Drawn with complete accuracy a parabola through these points will peak between x = 3 in and x = 7, though unless you have a very fine sense of the shape of a parabola your sketch might well peak somewhere to the right of x = 7. The peak of the actual parabola will occur close to x = 6, and the value at x = 7 will be just a bit greater than 8, perhaps 8.5 or so. If your peak was to the right of x = 7, your x = 5 value will be lass than 7.
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RESPONSE --> I understand the graph.
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20:44:34 `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.
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RESPONSE --> The x hits at approximately 1.5 when you follow the y axis 0.
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20:51:35 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).
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RESPONSE --> I estimated for the x coordinate instead of the y. Now that I see the answer I understand what you did. But where is the 0 y axis?
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21:04:13 `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?
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RESPONSE --> If we use the previous graph, we will have approx. 6 cents of profit for selling an item at $4. If our profit is 7 cents, the estimated selling price is $5.50 This is not realistic because you will have fluctuations and variables that are not represented on the graph, and it is not realistic to increase the price by $1.50 and only make 1 cent more profit. Furtehrmore if you choose another point say solve for 3 cents profit, you will have to have a selling price of approx. $2.50, which does not coorespond with the first estimate of $4 = 6 cents profit. I don't know what we are selliing, but we need to go into a different business if our profit is cents and selling is dollars. It should be the other way around, or we are headed for bankruptcy court.
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21:05:39 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. 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 (or x = 7.7, approx.).
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RESPONSE --> ok.
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21:11:10 `q007. On another set of coordinate axes, plot the points (-3, 4) and (5, -2). Sketch a straight line through these points. We will 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?
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RESPONSE --> Given this info the equation will look like this: 4 = m (-3) +b for the first set and -2 = m (5) +b for the second set
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21:11:24 Substituting x = -3 and y = 4 into the form y = m x + b, we obtain the equation 4 = -3 m + b. Reversing the sides we have -3 m + b = 4.
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RESPONSE --> I got that answer
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21:11:50 `q008. Substitute the coordinates of the point (5, -2) into the form y = m x + b. What equation do you get?
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RESPONSE --> I get -2 = m (5) +b
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21:11:56 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
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RESPONSE --> i got it
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21:15:59 `q009. You have obtained the equations -3 m + b = 4 and 5 m + b = -2
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RESPONSE --> Enter, as appropriate, an answer to the question, a critique of your answer in response to a given answer, your insights regarding the situation at this point, notes to yourself, or just an OK.ok
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21:22:07 . Use the method of elimination to solve these simultaneous equations for m and b.
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RESPONSE --> -3m + b=4 -5m +b=-2 I mulitplied the first set by -1 3m -b = -4 -5m + b = -2 I eliminated the b's and solved for m -2m = -6 m = 3 Now plug in 3 -3 (3) +b =4 -9 +b = 4 add 9 to both sides b = 13 m= 3, b= 13
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21:38:14 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.
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RESPONSE --> I see why I got it wrong. If I have variables only, then I subtract the two equations. I understand the rest. However, I did check my math with my answer and it checked out also.
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21:43:45 `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?
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RESPONSE --> y = -3/4x + 7/4 This is significant because the -3/4 will be the slope or (rise/run) 7/4 is the y intercept. So if you know these you can plot it on a graph.
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21:53:09 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).
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RESPONSE --> I get it.
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