Question 1

The matrix associated with the solution to a system of linear equations in x, y, and z is given. Write the solution to the
system, if it exists.
-$$\left[ \begin{array} { l l l | r } 
1 & 0 & 0 & 3 \
0 & 1 & 0 & - 14 \
0 & 0 & 1 & 1
\end{array} \right]$$
A) $x = 3 , y = - 14 , z = - 1$
B) $x = 3 , y = - 14 , z = 1$
C) $x = - 3 , y = - 14 , z = - 1$
D) $x = - 3 , y = - 14 , z = 1$

Accepted Answer

The given matrix is in reduced row echelon form, directly indicating the solutions for $x$, $y$, and $z$ as $x = 3$, $y = -14$, and $z = 1$.

Question 2

Write the augmented matrix associated with the system.
-$\left\{ \begin{array} { c } - 2 x + 2 y + 7 z = 57 \ 8 x - 2 y - 2 z = - 34 \ - 2 x + 7 y + 9 z = 81 \end{array} \right.$
A)
$\left[ \begin{array} { r r r | r } - 2 & 2 & 7 & 57 \ 8 & - 2 & - 2 & - 34 \ - 2 & 7 & 9 & 81 \end{array} \right]$
B)
$\left[ \begin{array} { r r r } - 2 & 2 & 7 \ 8 & - 2 & - 2 \ - 2 & 7 & 9 \end{array} \right]$
C)
$\left[ \begin{array} { r r r | r } 57 & 7 & - 2 & - 2 \ - 34 & - 2 & - 2 & 8 \ 81 & 9 & 7 & - 2 \end{array} \right]$

D)

$\left[ \begin{array} { r r r | r } - 2 & 8 & - 2 & 57 \ 2 & - 2 & 7 & - 34 \ 7 & - 2 & 9 & 81 \end{array} \right]$

Accepted Answer

The augmented matrix correctly represents the system of equations with the coefficients of $x$, $y$, and $z$ in the first three columns and the constants on the right side of the equation in the fourth column, separated by a line.

Question 3

A psychologist studying the effects of good nutrition on the behavior of rabbits feeds one group a combination of three foods, I, II, and III. Each of these foods contains three additives, A, B, and C, that are used in the study.
The table below gives the percent of each additive that is present in each food. If the diet being used requires
19)1 g per day of A, 8.94 g of B, and 14.02 g of C, find the number of grams of food II that should be used each
Day) Let z represent the number of grams of food III. $\begin{array}{l|l|l|l} 
& \text { Food I| Food II } & \text { Food III } \
\hline \text { Additive A } & 10 \% & 10 \% & 11 \% \
\text { Additive B } & 6 \% & 2 \% & 15 \% \
\text { Additive C } & 8 \% & 6 \% & 13 \%
\end{array}$

A) $2.7 \mathrm { z } - 60$
B) $1.8 z + 54$
C) $128 - 3.2 \mathrm { z }$
D) $2.1 z + 63$

Accepted Answer

The answer of A psychologist studying the effects of good...

Question 4

Write the augmented matrix associated with the system.
-$$\left\{ \begin{array} { r r } 
- 6 x + 5 y & = 6 \
2 y + & 3 z = - 6 \
- 9 z & = 7
\end{array} \right.$$
A)
$\left[ \begin{array} { r r r | r } - 6 & 5 & 0 & 6 \ 0 & 2 & 3 & - 6 \ 0 & 0 & - 9 & 7 \end{array} \right]$

B)
$\left[ \begin{array} { r r r | r } - 6 & 5 & 6 & 0 \ 2 & 3 & - 6 & 0 \ - 9 & 7 & 0 & 0 \end{array} \right]$

C)
$\left[ \begin{array} { r r r | r } - 6 & 5 & 0 & 6 \ 2 & 3 & 0 & - 6 \ 7 & 0 & 0 & 7 \end{array} \right]$

D)

$\left[ \begin{array} { r r r | r } - 6 & 5 & 6 & 0 \ 2 & 3 & 0 & - 6 \ - 9 & 0 & 0 & 7 \end{array} \right]$

Accepted Answer

The augmented matrix correctly represents the system of equations with coefficients of $x, y, z$ in the first three columns and the constants on the right side of the equation in the fourth column.

Question 5

A bank gives three loans totaling $330,000 to a development company for the purchase of three business properties. The largest loan is $38,000 more than the sum of the other two. Find the amount of the largest loan.
A) $192,000
B) $195,000
C) $184,000
D) $180,000

Accepted Answer

The answer of A bank gives three loans totaling $330,000...

Question 6

A bakery sells three types of cakes. The table below gives the number of cups of flour, cups of sugar, and eggs needed to produce each type of cake. 
 To fill its orders for cakes, the bakery used 72 cups of flour, 48 cups of sugar, and 51 eggs. How many cakes of each
Type were made?
A) 19 cake I, 8 cake II, 3 cake III
B) 41 cake I, 8 cake II, 12 cake III
C) 17 cake I, 8 cake II, 3 cake III
D) 3 cake I, 8 cake II, 17 cake III

Accepted Answer

The answer of A bakery sells three types of cakes....

Question 7

Write the augmented matrix associated with the system.
-$$\left\{ \begin{array} { l } 
7 x + 5 y + 4 z = 65 \
4 x + 8 y + 4 z = 56 \
9 x + 8 y + 2 z = 93
\end{array} \right.$$

A)$\left[ \begin{array} { l l l | l } 7 & 4 & 9 & 65 \ 5 & 8 & 8 & 56 \ 4 & 4 & 2 & 93 \end{array} \right]$.
B)$\left[ \begin{array} { l l l } 7 & 5 & 4 \ 4 & 8 & 4 \ 9 & 8 & 2 \end{array} \right]$
C)$\left[ \begin{array} { l l l | l } 7 & 5 & 4 & 65 \ 4 & 8 & 4 & 56 \ 9 & 8 & 2 & 93 \end{array} \right]$..
D)
$\left[ \begin{array} { l l l | l } 7 & 5 & 4 & 65 \ 4 & 8 & 4 & 56 \ 9 & 8 & 2 & 93 \end{array} \right]$

Accepted Answer

The answer of Write the augmented matrix associated with the...

Question 8

A trust account manager has $400,000 to be invested in three different accounts. The accounts pay 5%, 6%, and 10%, and the goal is to earn $32,000. Assuming that x dollars are invested at 5%, y dollars are invested at 6%,And z dollars are invested at 8%, what limits must there be on z so that all investment values are non-negative? 
A) $200,000 \leq z \leq 240,000$
B) $150,000 \leq \mathrm { z } \leq 180,000$
C) $210,000 \leq z \leq 250,000$
D) $150,000 \leq \mathrm { z } \leq 200,000$

Accepted Answer

Given the total investment is $400,000, and the goal is to earn $32,000, we can set up the equation based on the interest rates and total investment: $0.05x + 0.06y + 0.10z = 32,000$ and $x + y + z = 400,000$. To find the limits on $z$, we consider the scenarios where either $x$ or $y$ is 0, as these would provide the extreme cases for $z$. When $x = 0$ and $y = 0$, the maximum amount is invested in $z$ at 10% to reach the $32,000 goal, leading to $z = 320,000$. However, since $z$ cannot exceed the total investment minus the minimum required in other investments to maintain non-negativity, and considering the error in the provided interest rate for $z$ (it should be 10% as per the question stem, not 8%), the correct limits for $z$ must ensure all investments are non-negative and the total interest goal is met. The provided options do not directly align with the corrected premise due to the discrepancy in the interest rate for $z$, but the approach involves adjusting for the maximum and minimum amounts that can be invested in $z$ while meeting the interest goal and respecting the total investment limit.

Question 9

A trust account manager has $300,000 to be invested in three different accounts. The accounts pay 5%, 6%, and 8%, and the goal is to earn $18,000. Assuming that x dollars are invested at 5%, y dollars are invested at 6%, and
Z dollars are invested at 8%, find x in terms of z. 
A) $x = 2 z$
B) $x = z$
C) $x = 100,000 + z$
D) $x = 300,000 - 3 z$

Accepted Answer

The correct answer is A because the sum of the investments must be $300,000, so x + y + z = 300,000. The interest earned must be $18,000, so 0.05x + 0.06y + 0.08z = 18,000. Substituting the first equation into the second equation, we get 0.05(300,000 - y - z) + 0.06y + 0.08z = 18,000. Simplifying, we get 15,000 - 0.05y - 0.05z + 0.06y + 0.08z = 18,000. Combining like terms, we get 15,000 + 0.03z = 18,000. Subtracting 15,000 from both sides, we get 0.03z = 3,000. Dividing both sides by 0.03, we get z = 100,000. Substituting this value of z back into the first equation, we get x + y + 100,000 = 300,000. Subtracting 100,000 from both sides, we get x + y = 200,000. Since x and y are both positive, we can conclude that x = 2z.

Question 10

A basketball fieldhouse seats 15,000. Courtside seats sell for $8, endzone for $7, and balcony for $5. A full house earns $88,000 in ticket revenue. If half the courtside and balcony seats and all the endzone seats are sold, the
Total ticket revenue is $51,000. How many of each type of seat are there?
A) 4000 courtside, 3000 endzone, 8000 balcony
B) 4000 courtside, 1500 endzone, 9500 balcony
C) 3200 courtside, 1800 endzone, 10,000 balcony
D) 3000 courtside, 2000 endzone, 10,000 balcony

Accepted Answer

The total number of seats is 15,000, and the total revenue when the house is full is $88,000. When half the courtside and balcony seats are sold along with all the endzone seats, the revenue is $51,000. By setting up equations based on the prices of the tickets and the total revenue, and solving these equations, we find that the distribution that meets these conditions is 3000 courtside, 2000 endzone, and 10,000 balcony seats.

Question 11

Write the augmented matrix associated with the system.
-$$\left\{ \begin{array} { r } 
- 2 x - 2 z = - 6 \
- 2 y + 6 z = 10 \
9 x + 7 y + 8 z = 32
\end{array} \right.$$
A)
$\left[ \begin{array} { r r r } - 2 & 0 & - 2 \ 0 & - 2 & 6 \ 9 & 7 & 8 \end{array} \right]$

B)
$\left[ \begin{array} { r r r | r } - 2 & 0 & - 2 & - 6 \ 0 & - 2 & 6 & 10 \ 9 & 7 & 8 & 32 \end{array} \right]$
C)$\left[ \begin{array} { r r r | r } - 2 & -2 & 0 & - 6 \ 0 & - 2 & 6 & 10 \ 9 & 7 & 8 & 32 \end{array} \right]$
D)
$$\left[ \begin{array} { r r r | r } 
- 2 & - 2 & 0 & - 6 \
- 2 & 6 & 0 & 10 \
9 & 7 & 8 & 32
\end{array} \right] \quad \left[  \right]$$

Accepted Answer

The augmented matrix correctly represents the system of equations with coefficients and constants in their respective places, including the vertical line separating the coefficients from the constants.

Question 12

The Hoffman Trucking Company has an order for three products, A, B, and C, for delivery. The table below gives the volume in cubic feet, the weight in pounds, and the value for insurance in dollars for a unit of each of
The products. If one of the company's trucks can carry 38,000 cubic feet and 57,100 pounds, and is insured to
Carry $29,750, how many units of product A can be carried on the truck? Assume that z represents the number
Of units of product C. $$\begin{array} { l | l | l | l } 
& \text { Product A } & \text { Product B } & \text { Product C } \
\hline \text { Unit Volume } \left( \mathrm { ft } ^ { 3 } \right) & 24 & 20 & 50 \
\text { Weight (lbs) } & 36 & 30 & 75 \
\text { Value (dollars) } & 150 & 180 & 120
\end{array}$$
A)$140-3.5 z$
B) $4.5 \mathrm { z } - 18$
C) $135 - 4 \mathrm { z }$
D) There is no solution.

Accepted Answer

The answer of The Hoffman Trucking Company has an order...

Question 13

The matrix associated with the solution to a system of linear equations in x, y, and z is given. Write the solution to the
system, if it exists.
-$$\left[ \begin{array} { r r r | r } 
1 & 1 & 1 & - 4 \
1 & - 1 & 4 & - 26 \
5 & 1 & 1 & - 24
\end{array} \right]$$
A) $x = - 5 , y = 5 , z = - 4$
B) $x = - 4 , y = 5 , z = - 5$
C) $x = - 4 , y = - 5 , z = 5$
D) $x = - 5 , y = - 4 , z = 5$

Accepted Answer

To solve the system represented by the augmented matrix, we can use Gaussian elimination or back-substitution if the matrix is already in reduced row echelon form. The solution $x = -5, y = 5, z = -4$ satisfies the system of equations represented by the matrix, indicating that the operations to reduce the matrix would lead to these values for $x$, $y$, and $z$.

Question 14

A young woman wins $150,000 and buys 5 new cars. She chooses from cars that are priced at $40,000, $30,000, and $25,000 and spends all of the money. How many $25,000 cars did she buy?
A) 0 or 2
B) 0, 1, 2, or 3
C) 6
D) 0

Accepted Answer

The woman spent exactly $150,000 on 5 cars. The only combination that fits this scenario with the given prices is buying 2 cars at $25,000 each and 3 cars at $30,000 each: (2 * $25,000) + (3 * $30,000) = $50,000 + $90,000 = $140,000, which is incorrect. Therefore, the correct combination is 0 cars at $25,000 and 5 cars at $30,000: (5 * $30,000) = $150,000.

Question 15

Write the augmented matrix associated with the system.
-$$\left\{ \begin{array} { r } 
6 x + 8 z = 64 \
4 y + 4 z = 24 \
6 x + 2 y + 3 z = 62
\end{array} \right.$$
A)
$\left[ \begin{array} { l l l | l } 6 & 8 & 0 & 64 \ 4 & 4 & 0 & 24 \ 6 & 2 & 3 & 62 \end{array} \right]$

B)

$\left[ \begin{array} { l l l } 6 & 0 & 8 \ 0 & 4 & 4 \ 6 & 2 & 3 \end{array} \right]$
C)
$\left[ \begin{array} { l l l | l } 6 & 0 & 8 & 64 \ 0 & 4 & 4 & 24 \ 6 & 2 & 3 & 62 \end{array} \right]$

D)

$\left[ \begin{array} { l l l | l } 6 & 0 & 6 & 64 \ 0 & 4 & 2 & 24 \ 8 & 4 & 3 & 62 \end{array} \right]$

Accepted Answer

The answer of Write the augmented matrix associated with the...

Question 16

A company offers three mutual fund plans for its employees. Plan I consists of 2 blocks of common stocks, 3 municipal bonds, and 4 blocks of preferred stock. Plan II consists of 2 blocks of common stocks, 2 municipal
Bonds, and 1 block of preferred stock. Plan III consists of 4 blocks of common stocks, 5 municipal bonds, and 5
Blocks of preferred stock. An employee combined these plans so that he has 22 blocks of common stock, 27
Municipal bonds, and 26 blocks of preferred stock. How many units of plan III might he have?
A) 0, 1, 2, 3, 4, or 5
B) 3, 4, 5, or 6
C) 0, 1, 2, 3, 4, 5, or 6
D) 3, 4, or 5

Accepted Answer

The employee has 22 blocks of common stock, 27 municipal bonds, and 26 blocks of preferred stock. Plan III consists of 4 blocks of common stocks, 5 municipal bonds, and 5 blocks of preferred stock. So, the employee could have 0, 1, 2, 3, 4, or 5 units of plan III.

Question 17

A man has a total of $110,000 invested in three rental properties. One property earns 7.5% per year on its investment, the second earns 9%, and the third earns 10%. The three properties earn a total of $9600 per year.
What is the largest amount that could be invested at 7.5%?
A) $56,000
B) $110,000
C) $50,000
D) $54,000

Accepted Answer

To maximize the amount invested at 7.5%, the remaining amounts should be invested at the higher rates to ensure the total income does not exceed $9600. Let x be the amount invested at 7.5%, y at 9%, and z at 10%. The total investment is $110,000, so x + y + z = $110,000. The total income from these investments is $9600, so 0.075x + 0.09y + 0.1z = $9600. To maximize x, minimize the amounts at higher interest rates, assuming they meet the total income requirement without exceeding it. If y and z are minimized while still achieving the $9600 total income, x can be maximized. Given the constraints, the largest amount that can be invested at 7.5% without the total income exceeding $9600 is $56,000.

Question 18

A trust account manager has $500,000 to be invested in three different accounts. The accounts pay 5%, 6%, and 8%, and the goal is to earn $34,000. Assuming that x dollars are invested at 5%, y dollars are invested at 6%, and Z dollars are invested at 8%, find y in terms of z. 
A) $y = - 600,000 + 2 z$
B) $\mathrm { y } = 900,000 - 3 \mathrm { z }$
C) $y = - 400,000 + 2 z$
D) $\mathrm { y } = 1,100,000 - 3 \mathrm { z }$

Accepted Answer

The total investment is $500,000, so $x + y + z = 500,000$. The interest goal is $34,000, so $0.05x + 0.06y + 0.08z = 34,000$. Solving these equations simultaneously, we find $y = 900,000 - 3z$.

Question 19

The matrix associated with the solution to a system of linear equations in x, y, and z is given. Write the solution to the
system, if it exists.
-$$\left[ \begin{array} { r r r | r } 
1 & - 1 & 4 & - 20 \
5 & 0 & 1 & - 4 \
1 & 3 & 1 & 8
\end{array} \right]$$
A) $x = 0 , y = 4 , z = - 4$
B) $x = - 4 , y = 4 , z = 0$
C) $x = 4 , y = 4 , z = - 4$
D) $x = - 4 , y = 0 , z = 4$

Accepted Answer

The solution to the system of linear equations is x = 0, y = 4, and z = -4.

Question 20

The matrix associated with the solution to a system of linear equations in x, y, and z is given. Write the solution to the
system, if it exists.
-$\left[ \begin{array} { r r r | r } 1 & 0 & 0 & - 5 \ 0 & 1 & 0 & 9 \ 0 & 0 & 1 & 9 \end{array} \right]$
A) $x = - 5 , y = 0 , z = 9$
B) $x = 9 , y = 9 , z = - 5$
C) $x = - 5 , y = 9 , z = 9$
D) $x = 0 , y = 9 , z = - 5$

Accepted Answer

The given matrix is in reduced row echelon form, directly showing the solutions for x, y, and z as -5, 9, and 9, respectively.

Quiz 7: Systems of Equations and Matrices

The matrix associated with the solution to a system of linear equations in x, y, and z is given. Write the solution to the system, if it exists. - $\left[ \begin{array} { l l l | r } 1 & 0 & 0 & 3 \\0 & 1 & 0 & - 14 \\0 & 0 & 1 & 1\end{array} \right]$

Write the augmented matrix associated with the system. - $\left\{ \begin{array} { c } - 2 x + 2 y + 7 z = 57 \\ 8 x - 2 y - 2 z = - 34 \\ - 2 x + 7 y + 9 z = 81 \end{array} \right.$

Write the augmented matrix associated with the system. - $\left\{ \begin{array} { r r } - 6 x + 5 y & = 6 \\2 y + & 3 z = - 6 \\- 9 z & = 7\end{array} \right.$

A bank gives three loans totaling $330,000 to a development company for the purchase of three business properties. The largest loan is $38,000 more than the sum of the other two. Find the amount of the largest loan.

A bakery sells three types of cakes. The table below gives the number of cups of flour, cups of sugar, and eggs needed to produce each type of cake. To fill its orders for cakes, the bakery used 72 cups of flour, 48 cups of sugar, and 51 eggs. How many cakes of each Type were made?

Write the augmented matrix associated with the system. - $\left\{ \begin{array} { l } 7 x + 5 y + 4 z = 65 \\4 x + 8 y + 4 z = 56 \\9 x + 8 y + 2 z = 93\end{array} \right.$

A trust account manager has $300,000 to be invested in three different accounts. The accounts pay 5%, 6%, and 8%, and the goal is to earn $18,000. Assuming that x dollars are invested at 5%, y dollars are invested at 6%, and Z dollars are invested at 8%, find x in terms of z.

Write the augmented matrix associated with the system. - $\left\{ \begin{array} { r } - 2 x - 2 z = - 6 \\- 2 y + 6 z = 10 \\9 x + 7 y + 8 z = 32\end{array} \right.$

The matrix associated with the solution to a system of linear equations in x, y, and z is given. Write the solution to the system, if it exists. - $\left[ \begin{array} { r r r | r } 1 & 1 & 1 & - 4 \\1 & - 1 & 4 & - 26 \\5 & 1 & 1 & - 24\end{array} \right]$

A young woman wins $150,000 and buys 5 new cars. She chooses from cars that are priced at $40,000, $30,000, and $25,000 and spends all of the money. How many $25,000 cars did she buy?

Write the augmented matrix associated with the system. - $\left\{ \begin{array} { r } 6 x + 8 z = 64 \\4 y + 4 z = 24 \\6 x + 2 y + 3 z = 62\end{array} \right.$

A man has a total of $110,000 invested in three rental properties. One property earns 7.5% per year on its investment, the second earns 9%, and the third earns 10%. The three properties earn a total of $9600 per year. What is the largest amount that could be invested at 7.5%?

A trust account manager has $500,000 to be invested in three different accounts. The accounts pay 5%, 6%, and 8%, and the goal is to earn $34,000. Assuming that x dollars are invested at 5%, y dollars are invested at 6%, and Z dollars are invested at 8%, find y in terms of z.

The matrix associated with the solution to a system of linear equations in x, y, and z is given. Write the solution to the system, if it exists. - $\left[ \begin{array} { r r r | r } 1 & - 1 & 4 & - 20 \\5 & 0 & 1 & - 4 \\1 & 3 & 1 & 8\end{array} \right]$

The matrix associated with the solution to a system of linear equations in x, y, and z is given. Write the solution to the system, if it exists. - $\left[ \begin{array} { r r r | r } 1 & 0 & 0 & - 5 \\ 0 & 1 & 0 & 9 \\ 0 & 0 & 1 & 9 \end{array} \right]$