Question 1

Solve the problem.
-Two cars leave a city and head in the same direction. After 7 hours, the faster car is 28 miles ahead of the slower car. The slower car has traveled 343 miles. Find the speeds of the two cars.
A)45 mph and 49 mph
B)49 mph and 53 mph
C)40 mph and 44 mph
D)51 mph and 55 mph

Accepted Answer

Let the speed of the slower car be x mph.
Let the speed of the faster car be y mph.

We know that the slower car travels 343 miles in 7 hours. Therefore:

x = 343/7
x = 49 mph

We also know that after 7 hours, the faster car is 28 miles ahead of the slower car. This means that in 7 hours, the faster car has traveled 343 + 28 = 371 miles. Therefore:

y = 371/7
y = 53 mph

Therefore, the speeds of the two cars are 49 mph and 53 mph. So, the best choice is B. Let the speed of the slower car be x mph
Then, the speed of the faster car = (x+28)/7 mph (since in 7 hours, the faster car is 28 miles ahead of the slower car)
Distance travelled by the slower car = 343 miles
Using the formula distance = speed × time, we get:
343 = x × 7
x = 49 mph
Therefore, the speed of the faster car = (49+28)/7 = 77/7 = 11 mph
Hence, the speeds of the two cars are 49 mph and 11 mph. Therefore, option B) is the correct answer.

Question 2

Solve the system using matrices.
-$$\begin{array} { c } 
x + 7 y = 51 \
- 4 x + 6 y = 0
\end{array}$$
A)(8, 7)
B)(-9, 7)
C)(9, 6)
D)dependent system

Accepted Answer

The system of equations can be represented by the matrix equation AX = B, where A = $\begin{bmatrix} 1 & 7 \ -4 & 6 \end{bmatrix}$, X = $\begin{bmatrix} x \ y \end{bmatrix}$, and B = $\begin{bmatrix} 51 \ 0 \end{bmatrix}$. Solving for X using the inverse of A, we find that x = 9 and y = 6.

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 measuree $$\frac { 3 } { 5 }$$ the measure of the second smallest angle. The largest angle is 9° less than eleven times the measure of the second smallest angle. Determine the measure of each angle.
A)smallest angle: 3°
B)smallest angle: 18° second smallest angle: 15° second smallest angle: 30°
Largest angle: 162° largest angle: 132°
C)smallest angle: 7.2°
D)smallest angle: 9° second smallest angle: 12° second smallest angle: 15°
Largest angle: 160.8° largest angle: 156°

Accepted Answer

The smallest angle is 9°, the second smallest angle is 15°, and the largest angle is 156°. The relationships given in the problem are satisfied: 9° is $\frac{3}{5}$ of 15°, and 156° is 9° less than eleven times 15°. The sum of the angles is 180°.

Question 4

Solve the system of equations using determinants.
-$$\begin{array} { l } 
3 x + 3 y = 45 \
2 x - 3 y = - 15
\end{array}$$
A)(-6, -9)
B)(-9, 6)
C)(9, 6)
D)(6, 9)

Accepted Answer

The answer of Solve the system of equations using determinants.
-\[\begin{array}...

Question 5

Solve the system using matrices.
-$$\begin{array} { l } 
5 x + 4 y = 32 \
- 9 x + 7 y = - 86
\end{array}$$
A)(7, -1)
B)(8, -2)
C)(-8, -1)
D)inconsistent system

Accepted Answer

The solution to the system of equations using matrices is (8, -2).

Question 6

Solve the system using matrices.
-$$\begin{array} { l } 
5 x - 4 y - z = - 3 \
x - 2 y - 2 z = - 24 \
- 8 x + y + z = - 48
\end{array}$$
A)(8, 7, 9)
B)(8, 9, 7)
C)(-8, 9, 16)
D)inconsistent system

Accepted Answer

To solve the system using matrices, we can write the system in matrix form as AX = B, where A is the coefficient matrix, X is the column matrix of variables, and B is the constant matrix. Solving this using matrix operations, we find that the solution is (8, 9, 7).

Question 7

Solve the system using matrices.
-$$\begin{array} { l } 
- 3 x - y - 2 z = - 38 \
9 x + 9 z = 117 \
2 y + z = 16
\end{array}$$
A)(-7, 5, 14)
B)(7, 6, 5)
C)(7, 5, 6)
D)inconsistent

Accepted Answer

The system of equations can be solved using matrix methods such as Gaussian elimination or matrix inversion. Solving the system, we find that the solution is x = 7, y = 5, z = 6, which corresponds to choice C.

Question 8

Solve the problem.
-Ms. Adams received a bonus check for $15,000. She decided to divide the money among three different investments. With some of the money, she purchased a municipal bond paying 5.5% simple interest. She
Invested twice the amount she paid for the municipal bond in a certificate of deposit paying 4.4% simple
Interest. Ms. Adams placed the balance of the money in a money market account paying 3.1% simple interest. If
Ms. Adams' total interest for one year was $540, how much was placed in each account?
A)municipal bond: $1250
B)municipal bond: $2000 certificate of deposit: $2500 certificate of deposit: $4000
Money market: $11,250 money market: $9000
C)municipal bond: $1000
D)municipal bond: $1500 certificate of deposit: $2000 certificate of deposit: $3000
Money market: $12,000 money market: $10,500

Accepted Answer

Let the amount invested in the municipal bond be $x$. Therefore, the amount invested in the certificate of deposit is $2x$, and the remaining amount, $15000 - x - 2x = 15000 - 3x$, is invested in the money market account. The total interest from all investments is $540, which can be represented as $0.055x + 0.044(2x) + 0.031(15000 - 3x) = 540$. Solving this equation for $x$ gives $x = 1500$, which means $2x = 3000$ for the certificate of deposit, and $15000 - 3x = 10500$ for the money market account.

Question 9

Solve the system using matrices.
-$$\begin{array} { l } 
4 x + y = 10 \
- 4 y + 3 z = 20 \
8 x - 2 y + 3 z = 40
\end{array}$$
A)(3, -2, 5)
B)(2, -2, 4)
C)dependent system
D)inconsistent system

Accepted Answer

The answer of Solve the system using matrices.
-\[\begin{array} { l...

Question 10

Evaluate the determinant.
-$$\left| \begin{array} { l l } 
9 & 2 \
4 & 7
\end{array} \right|$$
A)55
B)-55
C)71
D)-10

Accepted Answer

The determinant of a 2x2 matrix $\left| \begin{array} { l l } a & b \ c & d \end{array} \right|$ is calculated as $ad - bc$. For this matrix, it is $9*7 - 2*4 = 63 - 8 = 55$.

Question 11

Evaluate the determinant.
-$$\left| \begin{array} { c c } 
\frac { 1 } { 5 } & - \frac { 1 } { 10 } \
10 & 5
\end{array} \right|$$
A)-2
B)0
C)2
D) $\frac { 5 } { 2 }$

Accepted Answer

The determinant of a 2x2 matrix $\left| \begin{array} { c c } a & b \ c & d \end{array} \right|$ is calculated as $ad - bc$. For this matrix, $a = \frac{1}{5}$, $b = -\frac{1}{10}$, $c = 10$, and $d = 5$. Thus, the determinant is $\frac{1}{5} \cdot 5 - (-\frac{1}{10} \cdot 10) = 1 + 1 = 2$.

Question 12

Solve the problem.
-The Little Town Arts Center charges $22 for adults, $18 for senior citizens, and $7 for children under 12 for their live performances on Sunday afternoon. This past Sunday, the paid revenue was $11,615 for 755 tickets sold.
There were 44 more children than adults. How many children attended?
A)257 children
B)285 children
C)213 children
D)247 children

Accepted Answer

The total revenue equation is $22A + $18S + $7C = $11,615, and the total tickets equation is A + S + C = 755. Given that there were 44 more children than adults (C = A + 44), we can solve these equations simultaneously to find the number of children. Solving these equations, we find that there were 257 children.

Question 13

Solve the system of equations using determinants.
-$$\begin{array} { l } 
2 x - 2 y = - 2 \
2 y + 4 x = 20
\end{array}$$
A)(3, 4)
B)(-4, 3)
C)(4, 3)
D)infinite number of solutions

Accepted Answer

The answer of Solve the system of equations using determinants.
-\[\begin{array}...

Question 14

Solve the system using matrices.
-$$\begin{array} { l } 
x + y + z = - 4 \
x - y + 4 z = 19 \
2 x + 2 y + 2 z = - 9
\end{array}$$
A)(5, -4, -5)
B)(5, -5, -4)
C)dependent system
D)inconsistent system

Accepted Answer

The system is inconsistent because the third equation is a multiple of the first equation, but with a different constant term, leading to a contradiction.

Question 15

Solve the system of equations using determinants.
-$$\begin{array} { l } 
6 x + 5 y = - 15 \
5 x + y = - 22
\end{array}$$
A)(3, -5)
B)(-5, 3)
C)(-3, -5)
D)no solution

Accepted Answer

The answer of Solve the system of equations using determinants.
-\[\begin{array}...

Question 16

Evaluate the determinant.
-$$\left| \begin{array} { r r } 
3 & 2 \
- 1 & 1
\end{array} \right|$$
A)-5
B)1
C)5
D)7

Accepted Answer

The determinant of a 2x2 matrix $\left| \begin{array} { r r } a & b \ c & d \end{array} \right|$ is calculated as $ad - bc$. For this matrix, it is $3*1 - 2*(-1) = 3 + 2 = 5$.

Question 17

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 8 hours to fire. A platter takes 15 hours to prepare, 3 hours
To paint, and 4 hours to fire. A bread basket takes 4 hours to prepare, 14 hours to paint, and 7 hours to fire. If the
Workshop has 101 hours for prep time, 55 hours for painting, and 82 hours for firing, how many of each can be
Made?
A)2 serving bowls, 6 platters, 5 bread baskets
B)7 serving bowls, 6 platters, 3 bread baskets
C)6 serving bowls, 5 platters, 2 bread baskets
D)5 serving bowls, 2 platters, 6 bread baskets

Accepted Answer

We can create a system of equations based on the time needed for each type of product:

3x + 15y + 4z = 101 (prep time)
2x + 3y + 14z = 55 (painting time)
8x + 4y + 7z = 82 (firing time)

Solving this system of equations using matrix operations or substitution will give us x = 6, y = 5, and z = 2. This means we can make 6 serving bowls, 5 platters, and 2 bread baskets. Therefore, the best choice is C.

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 90° 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.
A)First angle is 30°, second angle is 105°, third angle is 45°
B)First angle is 28°, second angle is 105°, third angle is 43°
C)First angle is 34°, second angle is 107°, third angle is 39°
D)First angle is 36°, second angle is 101°, third angle is 43°

Accepted Answer

Given the conditions, let's denote the first angle as $A$, the second angle as $B$, and the third angle as $C$. The conditions can be translated into the following equations:1. $A + B = C + 90$2. $B - C = 2A$3. $A + B + C = 180$ (since the sum of angles in a triangle is 180°)Solving these equations simultaneously, we find that $A = 30°$, $B = 105°$, and $C = 45°$, which matches option A.

Question 19

Solve the system using matrices.
-$$\begin{array} { l } 
3 x + 4 y = 10 \
4 x - 2 y = - 16
\end{array}$$
A)(4, -2)
B)(-2, 4)
C)(-2, -4)
D)inconsistent system

Accepted Answer

The solution to the system of equations can be found by using matrix methods such as Gaussian elimination or inverse matrices. Solving the system, we find that $x = -2$ and $y = 4$.

Question 20

Evaluate the determinant.
-$$\left| \begin{array} { l l } 
- 6 & 5 \
- 2 & 2
\end{array} \right|$$
A)-2
B)-26
C)-22
D)2

Accepted Answer

The determinant of a 2x2 matrix $\left| \begin{array} { l l } a & b \ c & d \end{array} \right|$ is calculated as $ad - bc$. For this matrix, it is $(-6)(2) - (5)(-2) = -12 + 10 = -2$.

Quiz 4: Systems of Equations and Inequalities

Solve the problem. -Two cars leave a city and head in the same direction. After 7 hours, the faster car is 28 miles ahead of the slower car. The slower car has traveled 343 miles. Find the speeds of the two cars.

Solve the system using matrices. - $\begin{array} { c } x + 7 y = 51 \\- 4 x + 6 y = 0\end{array}$

Solve the system of equations using determinants. - $\begin{array} { l } 3 x + 3 y = 45 \\2 x - 3 y = - 15\end{array}$

Solve the system using matrices. - $\begin{array} { l } 5 x + 4 y = 32 \\- 9 x + 7 y = - 86\end{array}$

Solve the system using matrices. - $\begin{array} { l } 5 x - 4 y - z = - 3 \\x - 2 y - 2 z = - 24 \\- 8 x + y + z = - 48\end{array}$

Solve the system using matrices. - $\begin{array} { l } - 3 x - y - 2 z = - 38 \\9 x + 9 z = 117 \\2 y + z = 16\end{array}$

Solve the system using matrices. - $\begin{array} { l } 4 x + y = 10 \\- 4 y + 3 z = 20 \\8 x - 2 y + 3 z = 40\end{array}$

Evaluate the determinant. - $\left| \begin{array} { l l } 9 & 2 \\4 & 7\end{array} \right|$

Evaluate the determinant. - $\left| \begin{array} { c c } \frac { 1 } { 5 } & - \frac { 1 } { 10 } \\10 & 5\end{array} \right|$

Solve the system of equations using determinants. - $\begin{array} { l } 2 x - 2 y = - 2 \\2 y + 4 x = 20\end{array}$

Solve the system using matrices. - $\begin{array} { l } x + y + z = - 4 \\x - y + 4 z = 19 \\2 x + 2 y + 2 z = - 9\end{array}$

Solve the system of equations using determinants. - $\begin{array} { l } 6 x + 5 y = - 15 \\5 x + y = - 22\end{array}$

Evaluate the determinant. - $\left| \begin{array} { r r } 3 & 2 \\- 1 & 1\end{array} \right|$

Solve the system using matrices. - $\begin{array} { l } 3 x + 4 y = 10 \\4 x - 2 y = - 16\end{array}$

Evaluate the determinant. - $\left| \begin{array} { l l } - 6 & 5 \\- 2 & 2\end{array} \right|$

Quiz 4: Systems of Equations and Inequalities

Solve the problem. -Two cars leave a city and head in the same direction. After 7 hours, the faster car is 28 miles ahead of the slower car. The slower car has traveled 343 miles. Find the speeds of the two cars.

Solve the system using matrices. - x+7y=51−4x+6y=0\begin{array} { c } x + 7 y = 51 \\- 4 x + 6 y = 0\end{array}x+7y=51−4x+6y=0​

Solve the system of equations using determinants. - 3x+3y=452x−3y=−15\begin{array} { l } 3 x + 3 y = 45 \\2 x - 3 y = - 15\end{array}3x+3y=452x−3y=−15​

Solve the system using matrices. - 5x+4y=32−9x+7y=−86\begin{array} { l } 5 x + 4 y = 32 \\- 9 x + 7 y = - 86\end{array}5x+4y=32−9x+7y=−86​

Solve the system using matrices. - 5x−4y−z=−3x−2y−2z=−24−8x+y+z=−48\begin{array} { l } 5 x - 4 y - z = - 3 \\x - 2 y - 2 z = - 24 \\- 8 x + y + z = - 48\end{array}5x−4y−z=−3x−2y−2z=−24−8x+y+z=−48​

Solve the system using matrices. - −3x−y−2z=−389x+9z=1172y+z=16\begin{array} { l } - 3 x - y - 2 z = - 38 \\9 x + 9 z = 117 \\2 y + z = 16\end{array}−3x−y−2z=−389x+9z=1172y+z=16​

Solve the system using matrices. - 4x+y=10−4y+3z=208x−2y+3z=40\begin{array} { l } 4 x + y = 10 \\- 4 y + 3 z = 20 \\8 x - 2 y + 3 z = 40\end{array}4x+y=10−4y+3z=208x−2y+3z=40​

Evaluate the determinant. - ∣9247∣\left| \begin{array} { l l } 9 & 2 \\4 & 7\end{array} \right|​94​27​​

Evaluate the determinant. - ∣15−110105∣\left| \begin{array} { c c } \frac { 1 } { 5 } & - \frac { 1 } { 10 } \\10 & 5\end{array} \right|​51​10​−101​5​​

Solve the system of equations using determinants. - 2x−2y=−22y+4x=20\begin{array} { l } 2 x - 2 y = - 2 \\2 y + 4 x = 20\end{array}2x−2y=−22y+4x=20​

Solve the system using matrices. - x+y+z=−4x−y+4z=192x+2y+2z=−9\begin{array} { l } x + y + z = - 4 \\x - y + 4 z = 19 \\2 x + 2 y + 2 z = - 9\end{array}x+y+z=−4x−y+4z=192x+2y+2z=−9​

Solve the system of equations using determinants. - 6x+5y=−155x+y=−22\begin{array} { l } 6 x + 5 y = - 15 \\5 x + y = - 22\end{array}6x+5y=−155x+y=−22​

Evaluate the determinant. - ∣32−11∣\left| \begin{array} { r r } 3 & 2 \\- 1 & 1\end{array} \right|​3−1​21​​

Solve the system using matrices. - 3x+4y=104x−2y=−16\begin{array} { l } 3 x + 4 y = 10 \\4 x - 2 y = - 16\end{array}3x+4y=104x−2y=−16​

Evaluate the determinant. - ∣−65−22∣\left| \begin{array} { l l } - 6 & 5 \\- 2 & 2\end{array} \right|​−6−2​52​​

Solve the system using matrices. - $\begin{array} { c } x + 7 y = 51 \\- 4 x + 6 y = 0\end{array}$

Solve the system of equations using determinants. - $\begin{array} { l } 3 x + 3 y = 45 \\2 x - 3 y = - 15\end{array}$

Solve the system using matrices. - $\begin{array} { l } 5 x + 4 y = 32 \\- 9 x + 7 y = - 86\end{array}$

Solve the system using matrices. - $\begin{array} { l } 5 x - 4 y - z = - 3 \\x - 2 y - 2 z = - 24 \\- 8 x + y + z = - 48\end{array}$

Solve the system using matrices. - $\begin{array} { l } - 3 x - y - 2 z = - 38 \\9 x + 9 z = 117 \\2 y + z = 16\end{array}$

Solve the system using matrices. - $\begin{array} { l } 4 x + y = 10 \\- 4 y + 3 z = 20 \\8 x - 2 y + 3 z = 40\end{array}$

Evaluate the determinant. - $\left| \begin{array} { l l } 9 & 2 \\4 & 7\end{array} \right|$

Evaluate the determinant. - $\left| \begin{array} { c c } \frac { 1 } { 5 } & - \frac { 1 } { 10 } \\10 & 5\end{array} \right|$

Solve the system of equations using determinants. - $\begin{array} { l } 2 x - 2 y = - 2 \\2 y + 4 x = 20\end{array}$

Solve the system using matrices. - $\begin{array} { l } x + y + z = - 4 \\x - y + 4 z = 19 \\2 x + 2 y + 2 z = - 9\end{array}$

Solve the system of equations using determinants. - $\begin{array} { l } 6 x + 5 y = - 15 \\5 x + y = - 22\end{array}$

Evaluate the determinant. - $\left| \begin{array} { r r } 3 & 2 \\- 1 & 1\end{array} \right|$

Solve the system using matrices. - $\begin{array} { l } 3 x + 4 y = 10 \\4 x - 2 y = - 16\end{array}$

Evaluate the determinant. - $\left| \begin{array} { l l } - 6 & 5 \\- 2 & 2\end{array} \right|$