#$&* course Mth 163
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Given Solution: The vertex of the 'blue' parabola is at the point (2, -1), the 'lowest' point on the parabola. The vertex of the 'purple' parabola is at the point (-2, 3), the 'highest' point on the parabola. Self-critique: ------------------------------------------------ Self-critique rating: Question `q002: What are the coordinates of the other two 'basic points' of each parabola? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your Solution: going from left to right the points of the purple parabola is (-3, 1) and (-1, 1). The points of the second blue parabola are (1, 0) and (3, 0). confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 3
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Given Solution: For the 'blue' parabola, the points (1, 0) and (3, 0) are two units apart, and lie on the same horizontal line. The horizontal line is the x axis. For the 'purple' parabola, the points (-3, 1) and (-1, 1) are two units apart. These points lie on the horizontal line where y = 2. Self-critique: ------------------------------------------------ Self-critique rating: Question `q003: For the first parabola, the one whose vertex is (2, 1), how far would we have to move to the right or the left, starting from the vertex, in order to be directly above or below another of its 'basic points'? How far would we then have to move in the vertical direction to reach that point? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your Solution: For the parabola with vertex (2,1) you would have to move one unit to the right or left to hit the next point and two unit up vertically to hit the next point. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 3
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Given Solution: For the parabola with vertex (2, 1), if we move 1 unit to the right or left we will be at the point (3, 1) or (1, 1), putting us directly below one of the other two basic points. If we then move 2 units upward, we will be at the point (3, 3) or (1, 3). So if we move 1 unit to the right or left, we need to move 2 units upward to get to another basic point. Self-critique: ------------------------------------------------ Self-critique rating: Question `q004: For each of the other two parabolas, how far would we have to move to the right or the left, starting from the vertex, in order to be directly above or below another of its 'basic points'? How far would we then have to move in the vertical direction to reach that point? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your Solution: For the purple parabola you would have to move one unit to the right or left to hit the next point at -1 and -3 and two units downward to hit the point at 1. For the blue parabola you would have to move one unit to the left or right to hit the next point and one unit upward to hit the point at 0. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 3
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Given Solution: For either of the other two parabolas, if we move 1 unit to the right or left we will be directly above or below one of its basic points (above in the case of the 'third' parabola, whose vertex is (-2, 3), below in the case of the 'second' parabola, whose vertex is (2, -1).) To get to the basic points of the 'third' parabola we will need to move 2 units downward. To get to the basic points of the 'second' parabola we will need to move 1 unit upward. Self-critique: ------------------------------------------------ Self-critique rating: Question `q005. Solve the following system of simultaneous linear equations: 3a + 3b = 9 6a + 5b = 16. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I will use the elimination method to solve this system. 3a is half of 6a so I wil multiply 3a+3b=9 by -2 which gives -6a - 6b= -18 now I can eliminate the a's and what is left of my systems is -6b= -18 and 5b= 16 these combine to create -b=-2 or b=2 now plug in b to either system to find a. 3a + 3(2)=9 becomes 3a+6=9 which becomes 3a=3 which means a=1 so a=1 and b=2 confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 3
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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. STUDENT QUESTION I got my answer in a very different way than the solution given. I have been trying to remember things from the classes I took a long time ago and came up with this answer. Is it alright to use this method? INSTRUCTOR RESPONSE Here is a synopsis of your solution: I'll first solve the first equation for a: 3a+3b=9 so a+b=3 so a=3-b. Now I'll substitute this expression for a into the second equation 6 a + 5 b = 16 Replacing a with 3 - b: 6(3-b)+5b=16 18-6b+5b=16 -b=-2 b=2 a = 3 - b so a=3 - 2 = 1 Substituting a = 1 and b = 2 into the two equations we get 3(1)+3(2)=9 so 9 = 9 6(1)+5(2)=16 so 16 = 16. The solution checks with the two equations. You have an excellent solution. The method you have used is performed correctly and is equally valid with the method used in the solutions. It is called the 'substitution method'. For these first few problems in this course the substitution method and the elimination method are equally efficient. However the elimination method is also important, and since elimination works better on most of the problems we'll be encountering in the near future, it is the method I use in the given solutions. You can use either method, as long as you know both. However you might find the given solutions easier to understand if you use the elimination method. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q006. Solve the following system of simultaneous linear equations using the method of elimination: 4a + 5b = 18 6a + 9b = 30. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 4 and 6 have a common number of 12 so I will multiply the first system 4a+5b=18 by 3 and the second system 6a+9b=30 by -2. lets do the first one, 3(4a+5b=18) is 12a+15b=54 the second becomes -2(6a+9b=30) is -12a-18b=-60. now eliminate the a's and add. 15b=54 and -18b=-60 become -3b=-6 which means b=2 Now plug in b to find a. 4a+5(2)=18 is 4a+10=18 is 4a=8 so a=2 as well. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 3
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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: `q007. If y = 5x + 8, then for what value of x will we have y = 13? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Since we have the y value substitute and fid x. 13=5x+8 becomes 5=5x so x=1 if y=13. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 3
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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: " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!