Deck 9: Prelude to Calculus

Question 1

Represent the given system of linear equations as a matrix. Use alphabetical order for the variables.
$\begin{aligned} 6 x-3 y & =-7 \\ 6 x-3 y & =21\end{aligned}$

Answer 1

$\left[\begin{array}{ccc}6 & -3 & -7 \\ 6 & -3 & 21\end{array}\right]$

Question 2

Represent the given system of linear equations as a matrix. Use alphabetical order for the variables.
$\begin{aligned} 6 x+6 y-9 z & =-6 \\ 4 x-\frac{2}{3} y+8 z & =4 \\ 8 x+4 y+\sqrt{10} z & =-9\end{aligned}$

Answer 2

$\left[\begin{array}{rrrr}6 & 6 & -9 & -6 \\ 4 & -\frac{2}{3} & 8 & 4 \\ 8 & 4 & \sqrt{10} & -9\end{array}\right]$

Question 3

Interpret the given matrix as a system of linear equations. Use x for the first variable, y for the second variable, and z for the third variable. $\left[\begin{array}{ccc}4 & -6 & 8 \\ -2 & \frac{6}{7} & 7\end{array}\right]$

A)
$\left\{\begin{array}{r}4 x-6 y+8 z \\ -2 x+\frac{6}{7} y+7 z\end{array}\right\}$

B)
$\left\{\begin{aligned} 4 & =6 x+8 y \\ -2 & =\frac{6}{7} x+7 y\end{aligned}\right\}$

C)
$\left\{\begin{aligned} 4 x-6 y & =8 z \\ -2 x+\frac{6}{7} y & =7 z\end{aligned}\right\}$

D)
$\left\{\begin{aligned} 4 x-6 y & =8 \\ -2 x+\frac{6}{7} y & =7\end{aligned}\right\}$

Answer 3

$\left\{\begin{aligned} 4 x-6 y & =8 \\ -2 x+\frac{6}{7} y & =7\end{aligned}\right\}$

Question 4

Interpret the given matrix as a system of linear equations. Use x for the first variable, y for the second variable, and z for the third variable.

Question 5

Interpret the given matrix as a system of linear equations. Use x for the first variable, y for the second variable, and z for the third variable.

Question 6

Use Gaussian elimination to find all solutions to the given system of equations. Work with matrices at least until the back substitution stage is reached. Give the exact answer.

Question 7

Use Gaussian elimination to find all solutions to the given system of equations. Work with matrices at least until the back substitution stage is reached.

Question 8

The solution of the following system of equations is given by x = 1, y = -1, z = 4.

Question 9

Find a number b such that the system of linear equations has no solutions. Give the exact answer.

Question 10

Find a number b < 49 such that the system of linear equations has infinitely many solutions.

Question 11

Find a number b such that the system of linear equations has no solutions. Give the exact answer.

Question 12

The system of linear equations has no solutions if and only if b = 5 or b = 0.

Question 13

The system of linear equations has infinitely many solutions.

Question 14

Find all solutions to the given system of equations.

Question 15

Find all solutions to the given system of equations.

A) (-2, 6)
B) (2, -6)
C) (6, -2)
D) (-6, 2)

Question 16

Find all solutions to the given system of equations.

Question 17

Determine the value of x in the given system of linear equations. $\begin{aligned} x+2 y-z & =4 \\ -2 x-2 y+2 z & =-3 \\ -3 x-2 y-4 z & =3\end{aligned}$

A) $\frac{5}{2}$

B) $-\frac{5}{2}$

C) $-\frac{12}{7}$

D) $\frac{12}{7}$

Question 18

Use Gaussian elimination to find all solutions to the given system of equations. Work directly with equations rather than matrices.

Question 19

Use Gaussian elimination to find all solutions to the given system of equations. Work directly with equations rather than matrices.

Question 20

The solution of the following system of equations is given by x = 1, y = -1, z = 4.

Question 21

Find a number b such that the system of linear equations has no solutions. Give the exact answer.

Question 22

Find a number b < 61 such that the system of linear equations has infinitely many solutions.

Question 23

Find a number b such that the system of linear equations has no solutions. Give the exact answer.

Question 24

The system of linear equations has no solutions if and only if b = 12 or b = 0.

Question 25

The system of linear equations has infinitely many solutions.

Question 26

Use Gaussian elimination to find all solutions to the given system of equations. Work directly with equations rather than matrices.

Question 27

Use Gaussian elimination to find all solutions to the given system of equations. Work directly with equations rather than matrices.

A) (3, 2)
B) (-5, 0)
C) (-3, 0)
D) (0, -3)

Question 28

Use Gaussian elimination to find all solutions to the given system of equations. Work directly with equations rather than matrices.

Question 29

Find all solutions to the given system of equations.

Question 30

Find all solutions to the given system of equations.

Question 31

An ad for a snack consisting of peanuts and raisins states that one serving of the regular snack contains 10 peanuts and 25 raisins and has 110 calories. The lite version of the snack consists of 5 peanuts and 30 raisins per serving and, according to the ad, has 97 calories. How many calories are in each peanut and each raisin?

Question 32

At an educational district's office, three types of employee wages are incorporated into the budget: specialists, managers, and directors. Employees of the same classification earn the same wage district wide. At one location, there are 15 specialists, 5 managers, and 2 directors with a total annual salary budget of $1,070,000. At another location, there are 15 specialists, 2 managers, and 1 director with a total annual salary budget of $771,000. A third location has 14 specialists, 1 manager, and 2 directors with a total annual salary budget of $667,000.