Question 1

Solve the system using matrices.
-x + y + z = -2 x - y + 3z = -14
5x + 5y + 5z = 6

Accepted Answer

The answer of Solve the system using matrices.
-x + y...

Question 2

Write the augmented matrix for the system.
-2x + 8y = 34 9x + 2y = 85

Accepted Answer

The augmented matrix for the given system is
$$\begin{bmatrix}2 & 8 & 34 \ 9 & 2 & 85\end{bmatrix}.$$ 
Thus, the correct option is (B). The augmented matrix for the system 2x + 8y = 34, 9x + 2y = 85 is:

[2 8 | 34
9 2 | 85]

The correct option is B, which matches this matrix.

Question 3

Solve the problem.
-The sum of the measures of the angles of a triangle is 180°. The smallest angle of the triangle has a measure  the measure of the second smallest angle. The largest angle is 18° less than three times the measure of the second smallest angle. Determine the measure of each angle.

Accepted Answer

The answer of Solve the problem.
-The sum of the measures...

Question 4

Write the augmented matrix for the system.
-8x + 9y = 35 7y = 21

Accepted Answer

The answer of Write the augmented matrix for the system.
-8x...

Question 5

Write the augmented matrix for the system.
-9x + 3z = 93 -2y + 8z = 16
5x + 5y + 2z = 93

Accepted Answer

The augmented matrix for the system is:

[9 0 3 | 93]
[0 -2 8 | 16]
[5 5 2 | 93]

The correct matrix is provided in choice B.

Question 6

Solve the system using matrices.
-3x + 2y = -3 5x + 8y = 9

Accepted Answer

To solve the system using matrices, we can write it in matrix form AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix. The solution is found by calculating X = A^(-1)B. For this system, the solution is (-3, 3).

Question 7

Write the augmented matrix for the system.
-6x + 4y + 7z = 97 -2x + 8y + 6z = 54
6x + 5y + 3z = 72

Accepted Answer

The augmented matrix for the system is:
$$\left[\begin{array}{ccc|c}
6 & 4 & 7 & 97 \
-2 & 8 & 6 & 54 \
6 & 5 & 3 & 72 \
\end{array}ight]$$

The correct answer is B, which matches the matrix above. The augmented matrix has the coefficients of the variables and the constants on the right-hand side. It can be obtained by writing the coefficients in a matrix and appending a column of constants. For the given system, the augmented matrix is:

[6   4   7   | 97]
[-2  8   6   | 54]
[6   5   3   | 72]

This matches choice B.

Question 8

Solve the problem.
-The Little Town Arts Center charges $24 for adults, $13 for senior citizens, and $5 for children under 12 for their live performances on Sunday afternoon. This past Sunday, the paid revenue was $12,328 for 907 tickets sold.
There were 45 more children than adults. How many children attended?

Accepted Answer

The answer of Solve the problem.
-The Little Town Arts Center...

Question 9

Solve the problem.
-The Little Town Fine Arts Center charges $25 per adult and $13 per senior citizen for its performances. On a recent weekend evening when 557 people paid admission, the total receipts were $9353. How many who paid
Were senior citizens?

Accepted Answer

The answer of Solve the problem.
-The Little Town Fine Arts...

Question 10

Solve the problem.
-A ceramics workshop makes serving bowls, platters, and bread baskets to sell at its Winter Festival. A serving bowl takes 3 hours to prepare, 2 hours to paint, and 10 hours to fire. A platter takes 17 hours to prepare, 3 hours
To paint, and 4 hours to fire. A bread basket takes 4 hours to prepare, 17 hours to paint, and 7 hours to fire. If the
Workshop has 94 hours for prep time, 58 hours for painting, and 90 hours for firing, how many of each can be
Made?

Accepted Answer

To find the answer, we need to create a system of equations and solve for the number of each item that can be made based on the given time constraints. Let x, y, and z represent the number of serving bowls, platters, and bread baskets made, respectively. Then, we have the following system of equations:
3x + 17y + 4z ≤ 94 (prep time)
2x + 3y + 17z ≤ 58 (painting time)
10x + 4y + 7z ≤ 90 (firing time)
We want to maximize the total number of items made, which is x + y + z. To solve for x, y, and z, we can use substitution or elimination. Using elimination, we can multiply the first equation by 4, the second equation by 17, and the third equation by 2 and add them together:
12x + 68y + 16z ≤ 376
34x + 51y + 289z ≤ 986
20x + 8y + 14z ≤ 180
Then, we can multiply the third equation by 17 and subtract it from the second equation:
14x + 47y + 231z ≤ 478
Subtracting twice the first equation from this, we get:
41y + 167z ≤ 290
Finally, we can see that the only integer solution to this inequality is y = 4 and z = 2. Substituting these values back into the third equation, we get x = 6. Therefore, the answer is C: 6 serving bowls, 4 platters, and 2 bread baskets can be made in the given time.

Question 11

Solve the system using matrices.
--5x + 2y - z = -29 x + 6y - 8z = 26
4x + y + z = 42

Accepted Answer

Writing the system of equations in matrix form:

$\left(\begin{array}{ccc|c} -5 & 2 & -1 & -29 \ 1 & 6 & -8 & 26 \ 4 & 1 & 1 & 42 \end{array}ight)$

Performing row operations:

$\left(\begin{array}{ccc|c} -5 & 2 & -1 & -29 \ 0 & 32 & -13 & 55 \ 0 & -7 & 5 & 163 \end{array}ight)$

$\left(\begin{array}{ccc|c} -5 & 2 & -1 & -29 \ 0 & 1 & -\frac{13}{32} & \frac{55}{32} \ 0 & -7 & 5 & 163 \end{array}ight)$

$\left(\begin{array}{ccc|c} -5 & 2 & -1 & -29 \ 0 & 1 & -\frac{13}{32} & \frac{55}{32} \ 0 & 0 & \frac{111}{32} & \frac{1261}{32} \end{array}ight)$

$\left(\begin{array}{ccc|c} -5 & 2 & -1 & -29 \ 0 & 1 & -\frac{13}{32} & \frac{55}{32} \ 0 & 0 & 1 & \frac{1131}{111} \end{array}ight)$

$\left(\begin{array}{ccc|c} -5 & 2 & 0 & -22 \ 0 & 1 & 0 & 7 \ 0 & 0 & 1 & \frac{1131}{111} \end{array}ight)$

$\left(\begin{array}{ccc|c} 1 & 0 & 0 & 8 \ 0 & 1 & 0 & 7 \ 0 & 0 & 1 & \frac{1131}{111} \end{array}ight)$

Therefore, the solution is $(x, y, z) = (8, 7, \frac{1131}{111})$, which matches choice B. We can write the system in matrix form as:
\begin{bmatrix}-5 & 2 & -1 \ 1 & 6 & -8 \ 4 & 1 & 1 \\end{bmatrix} \begin{bmatrix}x \ y \ z\end{bmatrix} = \begin{bmatrix}-29 \ 26 \ 42\end{bmatrix}

Using row reduction, we can find the row echelon form:

\begin{bmatrix}-5 & 2 & -1 & -29\ 1 & 6 & -8 & 26\ 4 & 1 & 1 & 42\end{bmatrix} $\xrightarrow[]{R_1 \leftrightarrow R_2}$ \begin{bmatrix}1 & 6 & -8 & 26\ -5 & 2 & -1 & -29\ 4 & 1 & 1 & 42\end{bmatrix} $\xrightarrow[]{R_2 + 5R_1 ightarrow R_2}$ \begin{bmatrix}1 & 6 & -8 & 26\ 0 & 32 & -41 & 121\ 4 & 1 & 1 & 42\end{bmatrix} $\xrightarrow[]{R_3 - 4R_1 ightarrow R_3}$ \begin{bmatrix}1 & 6 & -8 & 26\ 0 & 32 & -41 & 121\ 0 & -23 & 33 & 26\end{bmatrix} $\xrightarrow[]{R_2 + \frac{23}{32}R_3 ightarrow R_2}$ \begin{bmatrix}1 & 6 & -8 & 26\ 0 & 0 & -\frac{35}{4} & \frac{1357}{32}\ 0 & -23 & 33 & 26\end{bmatrix} $\xrightarrow[]{-\frac{4}{35}R_3 ightarrow R_3}$ \begin{bmatrix}1 & 6 & -8 & 26\ 0 & 0 & -\frac{35}{4} & \frac{1357}{32}\ 0 & 92/5 & -132/5 & -104/35\end{bmatrix} $\xrightarrow[]{-\frac{4}{35}R_2 ightarrow R_2}$ \begin{bmatrix}1 & 6 & -8 & 26\ 0 & 0 & 35 & -\frac{1357}{8}\ 0 & 92/5 & -132/5 & -104/35\end{bmatrix} $\xrightarrow[]{\frac{20}{23}R_3 + R_2 ightarrow R_2}$ \begin{bmatrix}1 & 6 & -8 & 26\ 0 & 0 & 0 & \frac{310}{23}\ 0 & 92/5 & -132/5 & -104/35\end{bmatrix} $\xrightarrow[]{\frac{5}{92}R_2 ightarrow R_2}$ \begin{bmatrix}1 & 6 & -8 & 26\ 0 & 0 & 0 & \frac{155}{46}\ 0 & 92/5 & -132/5 & -104/35\end{bmatrix}

The last row of the row echelon form implies that $0x + \frac{155}{46} = \frac{155}{46} = 0$, which is a contradiction. Therefore, the system is inconsistent and has no solution.

The only option among the given choices that corresponds to an inconsistent system is D.

Question 12

Solve the system using matrices.
-4x + y = 10 -4y + 4z = 4
4x + 5y - 4z = 6

Accepted Answer

The answer of Solve the system using matrices.
-4x + y...

Question 13

Solve the system using matrices.
-5x - y + 6z = 85 -8x - 8z = -128
4y + z = 15

Accepted Answer

Rewriting the system in matrix form:
$\begin{pmatrix}5 & -1 & 6\ -8 & 0 & -8\ 0 & 4 & 1\end{pmatrix}$ $\begin{pmatrix}x\y\z\end{pmatrix}$ $=$ $\begin{pmatrix}85\-128\15\end{pmatrix}$

Using row reduction (Gaussian elimination), we can simplify the matrix to row echelon form:

$\begin{pmatrix}5 & -1 & 6\ 0 & -8/5 & -16/5\ 0 & 0 & 219/40\end{pmatrix}$ $\begin{pmatrix}x\y\z\end{pmatrix}$ $=$ $\begin{pmatrix}85\-36\57/4\end{pmatrix}$

Solving for $z$ first, we get $z = 219/40$. Plugging that into the second equation, we get $y = 15/4 - (219/40)\cdot 4 = 2$. Finally, plugging in both $y$ and $z$ into the first equation, we get $x = (85 + 2 - (6\cdot 219/40))/5 = 9$. Thus, the solution is $(x,y,z) = (9, 2, 219/40)$. Rounded to the nearest integer, the answer is (9, 2, 2), which corresponds to choice C. We can write the system in matrix form as:

$\left[\begin{array}{ccc|c} 
5 & -1 & 6 & 85 \
-8 & 0 & -8 & -128 \
0 & 4 & 1 & 15
\end{array}ight]$

Using row operations, we can reduce the matrix to row echelon form:

$\left[\begin{array}{ccc|c} 
5 & -1 & 6 & 85 \
0 & 4 & 2 & 59 \
0 & 0 & 1 & 2
\end{array}ight]$

Now, we can use back-substitution to solve for the variables:

$z = 2$

$4y + 2(2) = 59 \Rightarrow y = 13$

$5x - (-13) + 6(2) = 85 \Rightarrow x = 9$

Therefore, the solution is $(x,y,z) = (9,13,2)$, which matches choice C.

Question 14

Perform the indicated row transformation and write the new matrix.
- Multiply the numbers in the first row by -2 and add the product to the second row.

Accepted Answer

Multiplying the numbers in the first row by -2 gives [-28,-14,12]. Adding this to the second row gives [0,-5,20] which is the second row in the new matrix. The other choices do not correctly carry out the row transformation.

Question 15

Solve the problem.
-Ms. Adams received a bonus check for $18,000. She decided to divide the money among three different investments. With some of the money, she purchased a municipal bond paying 5.6% simple interest. She
Invested twice the amount she paid for the municipal bond in a certificate of deposit paying 4.9% simple
Interest. Ms. Adams placed the balance of the money in a money market account paying 3.3% simple interest. If
Ms) Adams' total interest for one year was $759, how much was placed in each account?

Accepted Answer

The answer of Solve the problem.
-Ms. Adams received a bonus...

Question 16

Solve the problem.
-Two cars leave a city and head in the same direction. After 2 hours, the faster car is 22 miles ahead of the slower car. The slower car has traveled 84 miles. Find the speeds of the two cars.

Accepted Answer

Let the speed of the slower car be x mph and the speed of the faster car be y mph. 
After 2 hours, the slower car has traveled 2x miles and the faster car has traveled 2y miles. 
We are given that the faster car is 22 miles ahead of the slower car. 
Therefore, 2y = 2x + 22 
Simplifying, we get y = x + 11 
We are also given that the slower car has traveled 84 miles in 2 hours. 
Therefore, x = 42 mph. 
Substituting x in the equation y = x + 11, we get y = 53 mph. 
Thus, the speeds of the two cars are 42 mph and 53 mph. Let the speed of the slower car be x mph and the faster car be y mph.
After 2 hours, the distance traveled by the slower car is 2x + 84 and the distance traveled by the faster car is 2y.
We know that the faster car is 22 miles ahead of the slower car, so we can set up the equation: 2y - (2x + 84) = 22
Simplifying the equation, we get: 2y - 2x - 84 = 22
2y - 2x = 106
y - x = 53
We also know that the slower car traveled 84 miles in 2 hours, so we can set up another equation: 2x = 84
Solving for x, we get x = 42
Substituting x = 42 into y - x = 53, we get y = 95
Therefore, the speeds of the two cars are 42 mph and 95 mph, and the best choice is C.

Question 17

Perform the indicated row transformation and write the new matrix.
- Multiply the first row by

Accepted Answer

To perform the indicated row transformation, we need to multiply the first row by the scalar 11ebb63d_1b3d_6212_b851_3bea68d9d768_TB8475_11. This means every entry in the first row will be multiplied by this scalar. The resulting first row will be 11ebb63d_1b3d_6214_b851_ed4c0232ebd5_TB8475_11. Therefore, the best choice is B. To get the new matrix, we need to multiply the first row by the given scalar. This means multiplying each element in the first row by the scalar. In this case, we are not given the actual values in the matrix, but we can see that the only difference between the first row in the answer choices is in the fourth element, which is the only element in the first row that changes when we multiply by the scalar. Since the fourth element in the first row is given as "b851" in the problem and "ed4c0232ebd5" in choice B, that must be the correct answer.

Question 18

Solve the problem.
-A student is asked to construct a triangle that satisfies the following criteria. The sum of the first and second angles is 108° more than the measure of the third angle. If the measure of the third angle is subtracted from the
Measure of the second angle, the result is twice the measure of the first angle. Find the measure of each angle of
The triangle.

Accepted Answer

Let the first angle be x, the second angle be y, and the third angle be z. 
From the first condition, we have:

x + y = z + 108

From the second condition, we have:

y - z = 2x

We know that the sum of angles in a triangle is 180°, so we can write:

x + y + z = 180

Substituting the first and second equations into this equation, we get:

z + 108 + 2z + z = 180
4z = 72
z = 18

Substituting z = 18 into the first equation, we get:

x + y = 126

Substituting z = 18 into the second equation, we get:

y - 18 = 2x

Now we have two equations with two variables. Solving for x and y, we get:

x = 36, y = 108

Therefore, the measure of each angle of the triangle is: 
First angle = 36°
Second angle = 108°
Third angle = 36°

The only choice with these angles is C. Let x be the measure of the first angle, y be the measure of the second angle, and z be the measure of the third angle. 
From the problem, we have: 
x + y = z + 108  (equation 1)
y - z = 2x     (equation 2)
Also, we know that the sum of the angles in a triangle is 180 degrees. Thus: 
x + y + z = 180  (equation 3) 
Now we can solve for x, y, and z. 
From equation 1: 
y = z + 108 - x 
Substitute into equation 2: 
(z + 108 - x) - z = 2x 
Simplify: 
108 = 3x + z 
Substitute into equation 3: 
x + (z + 108 - x) + z = 180 
Simplify: 
2z + 108 = 180 
z = 36 
Substitute back into equation 1: 
x + y = 144 
Substitute z = 36 into equation 2: 
y - 36 = 2x 
y = 2x + 36 
Substitute y = 2x + 36 into x + y = 144: 
x + (2x + 36) = 144 
3x = 108 
x = 36 
Now we have: 
x = 36 
y = 2x + 36 = 108 
z = 36 
Thus, the measures of the angles are 36°, 108°, and 36° which matches choice C.

Question 19

Solve the problem.
-Marcus earned $5600 during his summer vacation. He invested part of the money at 9% and the rest at 7%. If he received a total of $464 in interest at the end of the year, how much was invested at 9%?

Accepted Answer

Let x be the amount invested at 9% and (5600-x) be the amount invested at 7%. 
Now we can set up an equation based on the given information:

0.09x + 0.07(5600-x) = 464

Simplifying this equation gives us:

0.09x + 392 - 0.07x = 464

0.02x = 72

x = 3600

Therefore, Marcus invested $3600 at 9% and $2000 at 7%. Thus, the answer is choice D, $3600.

Question 20

Solve the problem.
-Jancie has $180,000 to invest to obtain annual income. She wants some of it invested in safe Certificates of Deposit yielding 5%. The rest she wants to invest in AA bonds yielding 12% per year. How much should she
Invest in each to realize exactly $18,100 per year?

Accepted Answer

Let's call the amount invested in CDs "x" and the amount invested in AA bonds "y". We can set up two equations based on the given information:

0.05x + 0.12y = 18,100
x + y = 180,000

We can solve for one variable in terms of the other in the second equation, and substitute that into the first equation:

x = 180,000 - y
0.05(180,000 - y) + 0.12y = 18,100
9,000 - 0.05y + 0.12y = 18,100
0.07y = 9,100
y = 130,000

So Jancie should invest $130,000 in AA bonds and the remaining $50,000 in CDs. This will give her an annual income of:

0.05($50,000) + 0.12($130,000) = $6,500 + $15,600 = $18,100

Therefore, the best choice is D. Let x be the amount invested in AA bonds yielding 12%. Then the amount invested in CDs yielding 5% is 180,000 - x. The annual income from the AA bonds is 0.12x and the annual income from the CDs is 0.05(180,000 - x). We want the sum of these two incomes to be equal to $18,100 per year:
0.12x + 0.05(180,000 - x) = 18,100
0.12x + 9,000 - 0.05x = 18,100
0.07x = 9,100
x = 130,000
Therefore, Jancie should invest $130,000 in AA bonds yielding 12% and $50,000 in CDs yielding 5%. This will result in an annual income of:
0.12(130,000) + 0.05(50,000) = 15,600 + 2,500 = $18,100. 
This is the only choice that meets the given criteria.

Quiz 9: Systems of Linear Equations

Solve the system using matrices. -x + y + z = -2 x - y + 3z = -14 5x + 5y + 5z = 6

Write the augmented matrix for the system. -2x + 8y = 34 9x + 2y = 85

Solve the problem. -The sum of the measures of the angles of a triangle is 180°. The smallest angle of the triangle has a measure the measure of the second smallest angle. The largest angle is 18° less than three times the measure of the second smallest angle. Determine the measure of each angle.

Write the augmented matrix for the system. -8x + 9y = 35 7y = 21

Write the augmented matrix for the system. -9x + 3z = 93 -2y + 8z = 16 5x + 5y + 2z = 93

Solve the system using matrices. -3x + 2y = -3 5x + 8y = 9

Write the augmented matrix for the system. -6x + 4y + 7z = 97 -2x + 8y + 6z = 54 6x + 5y + 3z = 72

Solve the problem. -The Little Town Fine Arts Center charges $25 per adult and $13 per senior citizen for its performances. On a recent weekend evening when 557 people paid admission, the total receipts were $9353. How many who paid Were senior citizens?

Solve the system using matrices. --5x + 2y - z = -29 x + 6y - 8z = 26 4x + y + z = 42

Solve the system using matrices. -4x + y = 10 -4y + 4z = 4 4x + 5y - 4z = 6

Solve the system using matrices. -5x - y + 6z = 85 -8x - 8z = -128 4y + z = 15

Perform the indicated row transformation and write the new matrix. - Multiply the numbers in the first row by -2 and add the product to the second row.

Solve the problem. -Two cars leave a city and head in the same direction. After 2 hours, the faster car is 22 miles ahead of the slower car. The slower car has traveled 84 miles. Find the speeds of the two cars.

Perform the indicated row transformation and write the new matrix. - Multiply the first row by

Solve the problem. -Marcus earned $5600 during his summer vacation. He invested part of the money at 9% and the rest at 7%. If he received a total of $464 in interest at the end of the year, how much was invested at 9%?

Solve the problem. -Jancie has $180,000 to invest to obtain annual income. She wants some of it invested in safe Certificates of Deposit yielding 5%. The rest she wants to invest in AA bonds yielding 12% per year. How much should she Invest in each to realize exactly $18,100 per year?