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Figure 20-7 Illustration for Questions and Answers
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17-6 through 17-10
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Part Two 335 Answer 17-7
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They are the equations we derived in Answers 17-3 and 17-5: x = (1/2)y 4 and x = 3y 9
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Question 17-8
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What is the x-intercept of the line representing the equation x = (1/2)y 4
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Answer 17-8
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We can answer this straightaway, because the equation is in SI form with x as the dependent variable It s 4
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Question 17-9
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What is the x-intercept of the line representing the equation x = 3y 9
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Answer 17-9
Again, we can infer this from the SI equation having x as the dependent variable It s 9
Question 17-10
Based the known slopes of the lines, and on the point data shown in Fig 20-7, at what points would the lines representing the two-by-two linear system intersect the graph of the equation y = 2
Answer 17-10
The graph of the equation y = 2 would appear as a vertical line in Fig 20-7, parallel to the x axis and running through the point ( 2, 0) To reach this line from the point (2, 3), we can travel in the negative y direction by 4 units along either of our existing lines First, let s move along the line for x = (1/2)y 4 The slope is 1/2 That means if we go 4 units in the negative y direction, we must go 4/2, or 2, units in the negative x direction to stay on the line That will put us at the point ( 2, 5) Now let s move along the line for the equation x = 3y 9 The slope is 3 Therefore, if we go 4 units in the negative y direction, we must go 4 3, or 12, units in the negative x direction to stay on the line That will get us to the point ( 2, 15)
18
Question 18-1
Here is a set of equations that forms a three-by-three linear system: 4x = 8 + 4y + 4z 2y = 5 + x 5z 4z = 13 2x + y
336 Review Questions and Answers
How can we put the first of these equations into the form ax + by + c = d, where a, b, c, and d are constants
Answer 18-1
Here are the steps we can take, one at a time, starting with the original equation: 4x = 8 + 4y + 4z 4x 4y = 8 + 4z 4x 4y 4z = 8
Question 18-2
How can we put the second equation in Question 18-1 into the form ax + by + c = d, where a, b, c, and d are constants
Answer 18-2
Here are the steps we can take, one at a time, starting with the original equation: 2y = 5 + x 5z x + 2y = 5 5z x + 2y + 5z = 5
Question 18-3
How can we put the third equation in Question 18-1 into the form ax + by + c = d, where a, b, c, and d are constants
Answer 18-3
Here are the steps we can take, one at a time, starting with the original equation: 4z = 13 2x + y 2x + 4z = 13 + y 2x y + 4z = 13
Question 18-4
Based on the rearrangements in Answers 18-1 through 18-3, how can we state the three-bythree linear system from Question 18-1 now What strategies can we use to solve it
Answer 18-4
We can state the system by combining the final equations from Answers 18-1 through 18-3, like this: 4x 4y 4z = 8 x + 2y + 5z = 5 2x y + 4z = 13
Part Two 337
We can solve this system in many different ways In Chap 18, we learned to solve systems of this type by getting rid of one variable, resulting in a two-by-two linear system, solving that system, and then substituting back to solve for the variable we eliminated That s the method we ll use here We can get rid of any of the three variables by morphing and adding any two pairs of the three-variable equations
Question 18-5
How can we obtain a two-by-two linear system in x and z from the three-by-three system as stated in Answer 18-4, using the first two equations and then the second two
Answer 18-5
Here are the first two equations from the three-by-three linear system as stated in Answer 18-4: 4x 4y 4z = 8 x + 2y + 5z = 5 We can divide the top equation through by 2 and then add the bottom equation, getting the sum 2x 2y 2z = 4 x + 2y + 5z = 5 x + 3z = 9 That s the first equation in our two-by-two system To get the second equation, let s look at the second two equations from the three-by-three system as stated in Answer 18-4: x + 2y + 5z = 5 2x y + 4z = 13 We can multiply the bottom equation through by 2 and then add it to the top equation, getting x + 2y + 5z = 5 4x 2y + 8z = 26 3x + 13z = 31 Now we have the following two-by-two linear system in the variables x and z : x + 3z = 9 3x + 13z = 31
Question 18-6
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