Question 1

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form.
-$$\begin{array} { r } 
x - y + 5 z = - 21 \
5 x + z = - 4 \
x + 2 y + z = - 2
\end{array}$$

Accepted Answer

The answer of Use Gaussian elimination to solve the linear...

Question 2

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form.
-$$\begin{array} { l } 
2 x + 5 y + z = - 26 \
3 x - 3 y - z = 21 \
3 x + y + 3 z = - 11
\end{array}$$

Accepted Answer

By applying Gaussian elimination, the system is reduced to triangular form, leading to the solution (1, -5, -3).

Question 3

Solve the system of linear equations. If the system has infinitely many solutions, describe the solution with an ordered triple in terms of variable z.
-x + 8y + 8z = 8 
7x + 7y + z = 1 
8x + 15y + 9z = -9

Accepted Answer

The answer of Solve the system of linear equations. If...

Question 4

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form.
-$$\begin{array} { l } 
x + y + z = 0 \
x - y + 4 z = 0 \
2 x + y + z = 5
\end{array}$$

Accepted Answer

The system can be solved using Gaussian elimination to obtain a triangular form, leading to the solution (5, -3, -2).

Question 5

Solve the system of linear equations using the elimination method.
-$$\begin{aligned}
- 3 x - y - 6 z = & - 75 \
5 x + 5 y - 5 z = & 35 \
- 9 x - 7 y + z = & - 109
\end{aligned}$$

Accepted Answer

By using the elimination method, we can solve the system and find that the solution is (6, 9, 8).

Question 6

Solve the problem.
-A basketball player scored 24 points in a game. The number of three-point field goals the player made was 14 less than three times the number of free throws (each worth 1 point). Twice the number of two-point field goals the player made was 15 more than the number of three-point field goals made. Find the number of free-throws, two-point field goals, and three-point field goals that the player made in the game.

Accepted Answer

The player made 5 free throws, 8 two-point field goals, and 1 three-point field goal. This can be determined by setting up equations based on the information given and solving them.

Question 7

Solve the problem.
-The Little Town Arts Center charges $25 for adults, $14 for senior citizens, and $10 for children under 12 for their live performances on Sunday afternoon. This past Sunday, the paid revenue was $12,800 for 792 tickets sold. There were 41 more children than adults. How many children attended?

Accepted Answer

The total number of tickets sold was 792. Let's denote the number of adults as A, senior citizens as S, and children as C. We have three equations based on the information provided:1. $A + S + C = 792$ (total tickets)2. $25A + 14S + 10C = 12800$ (total revenue)3. $C = A + 41$ (41 more children than adults)From equation 3, we can express A in terms of C: $A = C - 41$.Substituting $A = C - 41$ into equation 1 gives us $C - 41 + S + C = 792$, which simplifies to $2C + S - 41 = 792$. However, to solve this system of equations directly, we need to manipulate these equations to isolate one variable.Given the complexity of solving this system with the information provided, we can approach the problem by testing the given options for the number of children and checking which one fits the conditions.Option D, 309 children, when tested with the conditions provided, will satisfy the equations when calculated correctly. This involves substituting the value back into the equations and solving for the other variables to ensure the total revenue matches. This is a more straightforward approach for multiple-choice questions when direct calculation might be too cumbersome without additional information or a mistake in the initial setup.

Question 8

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form.
-$$\begin{aligned}
x - y + 5 z & = - 5 \
3 x + z & = 0 \
x + 3 y + z & = 15
\end{aligned}$$

Accepted Answer

The answer of Use Gaussian elimination to solve the linear...

Question 9

Solve the system of linear equations. If the system has infinitely many solutions, describe the solution with an ordered triple in terms of variable z.
-$$\begin{array} { r r } 
x + y + z & = 7 \
x - y + 2 z & = 7 \
2 x + 3 z & = 14
\end{array}$$

Accepted Answer

The system of equations can be solved to express x and y in terms of z, resulting in the ordered triple $\left( - \frac { 3 z } { 2 } + 7 , \frac { z } { 2 } , z \right)$.

Question 10

Use Gaussian elimination and matrices to solve the system of linear equations. Write your final augmented matrix in triangular form and then solve for each variable using back substitution.
-$$\begin{aligned}
x + y + z & = 3 \
x - y + 2 z & = 5 \
4 x + y + z & = - 3
\end{aligned}$$

Accepted Answer

The system of equations can be represented as an augmented matrix and reduced to triangular form using Gaussian elimination. After performing the necessary row operations, the matrix can be reduced, and back substitution can be used to find the solution $x = -2$, $y = 1$, and $z = 4$.

Question 11

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form.
-$$\begin{array} { l } 
5 x + 4 y + z = - 16 \
5 x - 2 y - z = - 2 \
4 x + y + 3 z = 7
\end{array}$$

Accepted Answer

To solve the system using Gaussian elimination, we aim to transform the system into an upper triangular form. Starting with the given system:1. $5x + 4y + z = -16$2. $5x - 2y - z = -2$3. $4x + y + 3z = 7$First, eliminate $x$ from equations 2 and 3 using equation 1. To do this, we can subtract equation 1 from equation 2 and multiply equation 1 by $\frac{4}{5}$ and subtract it from equation 3 to eliminate $x$.After performing these operations, we get a new system (simplified for clarity):1. $5x + 4y + z = -16$2. $-6y - 2z = 14$3. $-3y + 2z = 15$Next, we eliminate $y$ from equation 3 using equation 2. We can multiply equation 2 by $\frac{1}{2}$ and add it to equation 3 to eliminate $y$.This results in:1. $5x + 4y + z = -16$2. $-6y - 2z = 14$3. $z = 5$Now, we can back-substitute $z = 5$ into the previous equations to find $y$ and $x$.Substituting $z = 5$ into equation 2:$-6y - 2(5) = 14$$-6y = 24$$y = -4$Finally, substitute $z = 5$ and $y = -4$ into equation 1:$5x + 4(-4) + 5 = -16$$5x - 16 + 5 = -16$$5x = -5$$x = -1$Thus, the solution to the system is $x = -1$, $y = -4$, $z = 5$, which corresponds to choice C.

Question 12

Solve the system of linear equations. If the system has infinitely many solutions, describe the solution with an ordered triple in terms of variable z.
-$$\begin{aligned}
5 x + 2 y + z & = - 11 \
2 x - 3 y - z & = 17 \
7 x - y & = 12
\end{aligned}$$

Accepted Answer

The answer of Solve the system of linear equations. If...

Question 13

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 9 hours to fire. A platter takes 16 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 119 hours for prep time, 84 hours for painting, and 122 hours for firing, how many of each can be made?

Accepted Answer

This problem can be solved using a system of linear equations based on the time constraints for each type of item. The equations are: 3S + 16P + 4B = 119 (prep time), 2S + 3P + 17B = 84 (painting time), and 9S + 4P + 7B = 122 (firing time), where S, P, and B represent the number of serving bowls, platters, and bread baskets, respectively. Solving this system yields the solution: 9 serving bowls, 5 platters, and 3 bread baskets.

Question 14

Solve the system of linear equations. If the system has infinitely many solutions, describe the solution with an ordered triple in terms of variable z.
-$$\begin{aligned}
x + 3 y + 2 z & = 11 \
4 y + 9 z & = - 12 \
x + 7 y + 11 z & = - 1
\end{aligned}$$

Accepted Answer

The answer of Solve the system of linear equations. If...

Question 15

Solve the problem.
-Find the values of $a$, $b$, and $c$ such that the graph of the quadratic equation $y = a x ^ { 2 } + b x + c$ passes through the points $( - 3,5 ) , ( 1,1 )$, and $( 2 , - 5 )$.

Accepted Answer

To find the values of $a$, $b$, and $c$, we substitute the given points into the quadratic equation $y = ax^2 + bx + c$ and solve the resulting system of equations. For $(-3, 5)$:$$5 = a(-3)^2 + b(-3) + c$$$$5 = 9a - 3b + c$$For $(1, 1)$:$$1 = a(1)^2 + b(1) + c$$$$1 = a + b + c$$For $(2, -5)$:$$-5 = a(2)^2 + b(2) + c$$$$-5 = 4a + 2b + c$$Solving this system of equations yields $a = -1$, $b = -3$, and $c = 5$, which corresponds to choice D.

Question 16

Solve the system of linear equations using the elimination method.
-x = -5 - y - z 
x - y + 4z = 2 
2x + y =-4 - z

Accepted Answer

The correct answer is A because it is the only choice that satisfies all three equations.

Question 17

Solve the system of linear equations. If the system has infinitely many solutions, describe the solution with an ordered triple in terms of variable z.
-$$\begin{aligned}
4 x - y + 3 z & = 12 \
x + 4 y + 6 z & = - 32 \
5 x + 3 y + 9 z & = 20
\end{aligned}$$

Accepted Answer

The answer of Solve the system of linear equations. If...

Question 18

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form.
-$$\begin{array} { r } 
x + y + z = - 10 \
x - y + 5 z = - 24 \
3 x + y + z = - 14
\end{array}$$

Accepted Answer

The answer of Use Gaussian elimination to solve the linear...

Question 19

Solve the problem.
-A deli sells three sizes of chicken sandwiches: the small chicken sandwich contains 4 ounces of meat and sells for $3.00; the regular chicken sandwich contains 8 ounces of meat and sells for $3.50; and the large chicken sandwich contains 10 ounces of meat and sells for $4.00. A customer requests a selection of each size for a reception. She and the manager agree on a combination of 52 sandwiches made from 22 pounds 4 ounces of chicken for a total cost of $178. How many of each size sandwich will be in this combination? (Note: 1 pound = 16 ounces)

Accepted Answer

The total weight in ounces is 22 pounds 4 ounces = (22 * 16) + 4 = 356 ounces. Let x, y, z represent the number of small, regular, and large sandwiches, respectively. We have three equations: 4x + 8y + 10z = 356 (total meat), x + y + z = 52 (total sandwiches), and 3x + 3.5y + 4z = 178 (total cost). Solving these equations gives us x = 22, y = 16, z = 14.

Question 20

Solve the system of linear equations. If the system has infinitely many solutions, describe the solution with an ordered triple in terms of variable z.
-$$\begin{aligned}
x + y + z & = 9 \
2 x - 3 y + 4 z & = 7 \
x - 4 y + 3 z & = - 2
\end{aligned}$$

Accepted Answer

The system of equations is consistent and dependent, leading to infinitely many solutions. Solving the system using substitution or elimination gives the solution in terms of z as $\left( - \frac { 7 z } { 5 } + \frac { 34 } { 5 } , \frac { 2 z } { 5 } + \frac { 11 } { 5 } , z \right)$.

Quiz 7: Systems of Equations and Inequalities

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form. - $\begin{array} { r } x - y + 5 z = - 21 \\5 x + z = - 4 \\x + 2 y + z = - 2\end{array}$

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form. - $\begin{array} { l } 2 x + 5 y + z = - 26 \\3 x - 3 y - z = 21 \\3 x + y + 3 z = - 11\end{array}$

Solve the system of linear equations. If the system has infinitely many solutions, describe the solution with an ordered triple in terms of variable z. -x + 8y + 8z = 8 7x + 7y + z = 1 8x + 15y + 9z = -9

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form. - $\begin{array} { l } x + y + z = 0 \\x - y + 4 z = 0 \\2 x + y + z = 5\end{array}$

Solve the system of linear equations using the elimination method. - $\begin{aligned}- 3 x - y - 6 z = & - 75 \\5 x + 5 y - 5 z = & 35 \\- 9 x - 7 y + z = & - 109\end{aligned}$

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form. - $\begin{aligned}x - y + 5 z & = - 5 \\3 x + z & = 0 \\x + 3 y + z & = 15\end{aligned}$

Solve the system of linear equations. If the system has infinitely many solutions, describe the solution with an ordered triple in terms of variable z. - $\begin{array} { r r } x + y + z & = 7 \\x - y + 2 z & = 7 \\2 x + 3 z & = 14\end{array}$

Use Gaussian elimination and matrices to solve the system of linear equations. Write your final augmented matrix in triangular form and then solve for each variable using back substitution. - $\begin{aligned}x + y + z & = 3 \\x - y + 2 z & = 5 \\4 x + y + z & = - 3\end{aligned}$

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form. - $\begin{array} { l } 5 x + 4 y + z = - 16 \\5 x - 2 y - z = - 2 \\4 x + y + 3 z = 7\end{array}$

Solve the system of linear equations. If the system has infinitely many solutions, describe the solution with an ordered triple in terms of variable z. - $\begin{aligned}5 x + 2 y + z & = - 11 \\2 x - 3 y - z & = 17 \\7 x - y & = 12\end{aligned}$

Solve the system of linear equations. If the system has infinitely many solutions, describe the solution with an ordered triple in terms of variable z. - $\begin{aligned}x + 3 y + 2 z & = 11 \\4 y + 9 z & = - 12 \\x + 7 y + 11 z & = - 1\end{aligned}$

Solve the problem. -Find the values of $a$ , $b$ , and $c$ such that the graph of the quadratic equation $y = a x ^ { 2 } + b x + c$ passes through the points $( - 3,5 ) , ( 1,1 )$ , and $( 2 , - 5 )$ .

Solve the system of linear equations using the elimination method. -x = -5 - y - z x - y + 4z = 2 2x + y =-4 - z

Solve the system of linear equations. If the system has infinitely many solutions, describe the solution with an ordered triple in terms of variable z. - $\begin{aligned}4 x - y + 3 z & = 12 \\x + 4 y + 6 z & = - 32 \\5 x + 3 y + 9 z & = 20\end{aligned}$

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form. - $\begin{array} { r } x + y + z = - 10 \\x - y + 5 z = - 24 \\3 x + y + z = - 14\end{array}$

Solve the system of linear equations. If the system has infinitely many solutions, describe the solution with an ordered triple in terms of variable z. - $\begin{aligned}x + y + z & = 9 \\2 x - 3 y + 4 z & = 7 \\x - 4 y + 3 z & = - 2\end{aligned}$

Quiz 7: Systems of Equations and Inequalities

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form. - x−y+5z=−215x+z=−4x+2y+z=−2\begin{array} { r } x - y + 5 z = - 21 \\5 x + z = - 4 \\x + 2 y + z = - 2\end{array}x−y+5z=−215x+z=−4x+2y+z=−2​

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form. - 2x+5y+z=−263x−3y−z=213x+y+3z=−11\begin{array} { l } 2 x + 5 y + z = - 26 \\3 x - 3 y - z = 21 \\3 x + y + 3 z = - 11\end{array}2x+5y+z=−263x−3y−z=213x+y+3z=−11​

Solve the system of linear equations. If the system has infinitely many solutions, describe the solution with an ordered triple in terms of variable z. -x + 8y + 8z = 8 7x + 7y + z = 1 8x + 15y + 9z = -9

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form. - x+y+z=0x−y+4z=02x+y+z=5\begin{array} { l } x + y + z = 0 \\x - y + 4 z = 0 \\2 x + y + z = 5\end{array}x+y+z=0x−y+4z=02x+y+z=5​

Solve the system of linear equations using the elimination method. - −3x−y−6z=−755x+5y−5z=35−9x−7y+z=−109\begin{aligned}- 3 x - y - 6 z = & - 75 \\5 x + 5 y - 5 z = & 35 \\- 9 x - 7 y + z = & - 109\end{aligned}−3x−y−6z=5x+5y−5z=−9x−7y+z=​−7535−109​

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form. - x−y+5z=−53x+z=0x+3y+z=15\begin{aligned}x - y + 5 z & = - 5 \\3 x + z & = 0 \\x + 3 y + z & = 15\end{aligned}x−y+5z3x+zx+3y+z​=−5=0=15​

Solve the system of linear equations. If the system has infinitely many solutions, describe the solution with an ordered triple in terms of variable z. - x+y+z=7x−y+2z=72x+3z=14\begin{array} { r r } x + y + z & = 7 \\x - y + 2 z & = 7 \\2 x + 3 z & = 14\end{array}x+y+zx−y+2z2x+3z​=7=7=14​

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form. - 5x+4y+z=−165x−2y−z=−24x+y+3z=7\begin{array} { l } 5 x + 4 y + z = - 16 \\5 x - 2 y - z = - 2 \\4 x + y + 3 z = 7\end{array}5x+4y+z=−165x−2y−z=−24x+y+3z=7​

Solve the problem. -Find the values of aaa , bbb , and ccc such that the graph of the quadratic equation y=ax2+bx+cy = a x ^ { 2 } + b x + cy=ax2+bx+c passes through the points (−3,5) ,(1,1) ( - 3,5 ) , ( 1,1 ) (−3,5) ,(1,1) , and (2,−5) ( 2 , - 5 ) (2,−5) .

Solve the system of linear equations using the elimination method. -x = -5 - y - z x - y + 4z = 2 2x + y =-4 - z

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form. - x+y+z=−10x−y+5z=−243x+y+z=−14\begin{array} { r } x + y + z = - 10 \\x - y + 5 z = - 24 \\3 x + y + z = - 14\end{array}x+y+z=−10x−y+5z=−243x+y+z=−14​

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form. - $\begin{array} { r } x - y + 5 z = - 21 \\5 x + z = - 4 \\x + 2 y + z = - 2\end{array}$

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form. - $\begin{array} { l } 2 x + 5 y + z = - 26 \\3 x - 3 y - z = 21 \\3 x + y + 3 z = - 11\end{array}$

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form. - $\begin{array} { l } x + y + z = 0 \\x - y + 4 z = 0 \\2 x + y + z = 5\end{array}$

Solve the system of linear equations using the elimination method. - $\begin{aligned}- 3 x - y - 6 z = & - 75 \\5 x + 5 y - 5 z = & 35 \\- 9 x - 7 y + z = & - 109\end{aligned}$

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form. - $\begin{aligned}x - y + 5 z & = - 5 \\3 x + z & = 0 \\x + 3 y + z & = 15\end{aligned}$

Solve the system of linear equations. If the system has infinitely many solutions, describe the solution with an ordered triple in terms of variable z. - $\begin{array} { r r } x + y + z & = 7 \\x - y + 2 z & = 7 \\2 x + 3 z & = 14\end{array}$

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form. - $\begin{array} { l } 5 x + 4 y + z = - 16 \\5 x - 2 y - z = - 2 \\4 x + y + 3 z = 7\end{array}$

Solve the problem. -Find the values of $a$ , $b$ , and $c$ such that the graph of the quadratic equation $y = a x ^ { 2 } + b x + c$ passes through the points $( - 3,5 ) , ( 1,1 )$ , and $( 2 , - 5 )$ .

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form. - $\begin{array} { r } x + y + z = - 10 \\x - y + 5 z = - 24 \\3 x + y + z = - 14\end{array}$