Questions: A professor wants to create a test that has 12 true-false questions, 8 multiple-choice questions, and 4 essay questions. She wants the test to be worth 200 points, with each multiple-choice question worth twice as many points as a true-false question and with each essay question equal to three times the number of points of a true-false question. a. Write a system of equations to represent this problem, with x, y, and z equal to the number of points for a true-false, multiple-choice, and essay question, respectively. Select all the equations that are needed for the system that represents this problem. A. 12x+8y+4z=0 B. 12x+8y+4z=200 C. x=2y D. z=3x E. z=2x F. x+y+z=0 G. x=2z H. x+y+z=200 I. y=2x J. y=3z

A professor wants to create a test that has 12 true-false questions, 8 multiple-choice questions, and 4 essay questions. She wants the test to be worth 200 points, with each multiple-choice question worth twice as many points as a true-false question and with each essay question equal to three times the number of points of a true-false question.
a. Write a system of equations to represent this problem, with x, y, and z equal to the number of points for a true-false, multiple-choice, and essay question, respectively. Select all the equations that are needed for the system that represents this problem.
A. 12x+8y+4z=0 B. 12x+8y+4z=200
C. x=2y D. z=3x
E. z=2x F. x+y+z=0
G. x=2z H. x+y+z=200
I. y=2x J. y=3z

Transcript text: A professor wants to create a test that has 12 true-false questions, 8 multiple-choice questions, and 4 essay questions. She wants the test to be worth 200 points, with each multiple-choice question worth twice as many points as a true-false question and with each essay question equal to three times the number of points of a true-false question. a. Write a system of equations to represent this problem, with $x, y$, and $z$ equal to the number of points for a true-false, multiple-choice, and essay question, respectively. Select all the equations that are needed for the system that represents this problem. A. $12 x+8 y+4 z=0$ B. $12 x+8 y+4 z=200$ C. $x=2 y$ D. $z=3 x$ E. $z=2 x$ F. $x+y+z=0$ G. $x=2 z$ H. $x+y+z=200$ I. $y=2 x$ J. $y=3 z$

Solution

Solution Steps

Step 1: Express y and z in terms of x

Given that each multiple-choice question is worth twice as many points as a true-false question, we have $y = 2x$. Similarly, since each essay question is worth three times as many points as a true-false question, we have $z = 3x$.

Step 2: Substitute y and z in terms of x into the equation $ax + by + cz = T$

Substituting $y = 2x$ and $z = 3x$ into the equation gives us $ax + b(2x) + c(3x) = T$. Simplifying, we get $x(a + 2b + 3c) = T$.

Step 3: Solve the resulting equation for x

Solving for $x$, we find that $x = \frac{T}{a + 2b + 3c} = 5$.

Step 4: Calculate y and z

Using the values of $x$, we calculate $y = 2x = 10$ and $z = 3x = 15$.