Question 1

Solve the system.
-$\left\{ \begin{array} { c } 
x - y + 2 z = 5 \
5 x + z = 0 \
- x + y - 2 z = - 10
\end{array} \right.$

Accepted Answer

The system is inconsistent. Adding the first and third equations gives $0 = -5$, which is a contradiction, indicating no solution exists.

Question 2

Without actually solving the problem, choose the correct solution by deciding which choice satisfies the given conditions.
-Aviva has a total of 70 coins, all of which are either dimes or nickels. The total value of the coins is \$5.95. Find the number of each type of coin.

Accepted Answer

The total value of the coins is $5.95, which is 595 cents. Since the problem involves nickels (5 cents) and dimes (10 cents), and we know the total number of coins is 70, we can set up an equation to represent this situation. However, without solving it directly, we can check which option gives us a total of 70 coins and adds up to 595 cents. Option B, with 21 nickels (21*5 = 105 cents) and 49 dimes (49*10 = 490 cents), adds up to 595 cents and totals 70 coins.

Question 3

Solve the system.
-$\left\{ \begin{array} { r } 
y + 5 z = - 8 \
- 4 x + y - 3 z = 8 \
2 x + 3 z = - 7
\end{array} \right.$

Accepted Answer

To solve the system, we can use substitution or elimination. Here's a brief way to solve it:1. From the third equation, $2x + 3z = -7$, we can express $x$ in terms of $z$: $x = \frac{-7 - 3z}{2}$.2. Substitute $x$ in the second equation: $-4\left(\frac{-7 - 3z}{2}\right) + y - 3z = 8$. Simplify and solve for $y$ in terms of $z$.3. Substitute $y$ and $x$ in terms of $z$ into the first equation and solve for $z$.Following these steps, we find $z = -1$. Substituting $z = -1$ back into the equations to find $x$ and $y$, we get $x = -2$ and $y = -3$. Therefore, the solution is $(-2, -3, -1)$, which corresponds to choice C.

Question 4

Solve the system.
-$\left\{ \begin{array} { r r } 
x - y + 5 z & = - 2 \
3 x + z & = 0 \
x + 3 y + z & = 6
\end{array} \right.$

Accepted Answer

The answer of Solve the system.
-\(\left\{ \begin{array} { r r...

Question 5

Write a system of equations in x and y describing the situation. Do not solve the system.
-An order of 2 orders of fries, 3 hamburgers, and 3 drinks costs $\$ 11$. An order of 4 orders of fries, 2 hamburgers, and 5 drinks costs $\$ 13$. All drinks are $\$ 1$.

Accepted Answer

The correct system of equations accounts for the cost of fries (x), hamburgers (y), and drinks (fixed at $1 each) to match the total cost of each order. Choices B, C, and D either incorrectly structure the equations or misrepresent the total costs.

Question 6

Write a system of equations in x and y describing the situation. Do not solve the system.
-One number is 5 more than another number. If you add 8 to 3 times the first number, the result is 4 times the second number.

Accepted Answer

The first equation $x - y = 5$ represents "one number is 5 more than another number." The second equation $3x - 4y = -8$ represents "if you add 8 to 3 times the first number, the result is 4 times the second number."

Question 7

Solve.
-The sum of two numbers is $- 5$. Three times the first number equals 4 times the second number. Find the two numbers.

Accepted Answer

Let the first number be $x$ and the second number be $y$. The system of equations based on the given information is:1) $x + y = -5$2) $3x = 4y$From equation 2, we can express $x$ in terms of $y$: $x = \frac{4}{3}y$.Substituting $x = \frac{4}{3}y$ into equation 1 gives us:$\frac{4}{3}y + y = -5$$\frac{7}{3}y = -5$$y = -\frac{15}{7}$Substituting $y = -\frac{15}{7}$ back into $x = \frac{4}{3}y$ gives us:$x = \frac{4}{3}(-\frac{15}{7}) = -\frac{20}{7}$Therefore, the two numbers are $-\frac{20}{7}$ and $-\frac{15}{7}$.

Question 8

Solve the system.
-$\left\{ \begin{aligned}
2 x + y - \frac { 3 } { 2 } z & = - 3 \
\frac { 3 } { 2 } x - 3 z & = 0 \
x - 3 y & = - 10
\end{aligned} \right.$

Accepted Answer

The answer of Solve the system.
-\(\left\{ \begin{aligned}
2 x + y...

Question 9

Solve the system.
-$\left\{ \begin{array} { r } 
x - y + 3 z = - 5 \
5 x + z = - 1 \
x + 5 y + z = 9
\end{array} \right.$

Accepted Answer

The answer of Solve the system.
-\(\left\{ \begin{array} { r }...

Question 10

Solve the system.
-$\left\{ \begin{array} { l } 
x - y + z = - 3 \
x + y + z = 7 \
x + y - z = - 1
\end{array} \right.$

Accepted Answer

Adding the first and third equations gives $2x = -4$, so $x = -2$. Substituting $x = -2$ into the second equation gives $-2 + y + z = 7$, so $y + z = 9$. Substituting $x = -2$ into the third equation gives $-2 + y - z = -1$, so $y - z = 1$. Solving these two equations for $y$ and $z$ gives $y = 5$ and $z = 4$. Therefore, the solution is $(-2, 5, 4)$.

Question 11

Solve the system.
-$\left\{ \begin{aligned}
x + y + z & = - 2 \
x - y + 4 z & = 14 \
4 x + 4 y + 4 z & = - 14
\end{aligned} \right.$

Accepted Answer

The third equation is equivalent to the first equation multiplied by 4, which means the system is dependent and does not have a unique solution. This inconsistency indicates there is no single solution set that satisfies all equations simultaneously.

Question 12

Solve the system.
-$\left\{ \begin{aligned}
x + 2 y + 4 z & = 0 \
5 y + 3 z & = 9 \
z & = - 2
\end{aligned} \right.$

Accepted Answer

Given $z = -2$, substituting into the second equation gives $5y + 3(-2) = 9$, which simplifies to $5y - 6 = 9$, leading to $5y = 15$, so $y = 3$. Substituting $y = 3$ and $z = -2$ into the first equation gives $x + 2(3) + 4(-2) = 0$, which simplifies to $x + 6 - 8 = 0$, leading to $x = 2$. Therefore, the solution is $(2, 3, -2)$.

Question 13

Solve the system.
-$\left\{ \begin{array} { r } 
x + y + z = - 10 \
x - y + 4 z = - 13 \
4 x + 4 y + 4 z = - 22
\end{array} \right.$

Accepted Answer

The third equation is equivalent to the first equation multiplied by 4, which means the system is dependent and does not have a unique solution.

Question 14

Without actually solving the problem, choose the correct solution by deciding which choice satisfies the given conditions.
-Trish has two containers of water for a camping trip. One container holds 2 times as much as the other. The total amount of water is 21 gallons. How much water is in each container?

Accepted Answer

The total amount of water is 21 gallons, and one container holds twice as much as the other. Only option C, with 7 gallons and 14 gallons, adds up to 21 gallons and satisfies the condition that one container holds twice as much as the other.

Question 15

Solve the system.
-$\left\{ \begin{array} { r r } 
x - 3 y - z = & - 5 \
5 x - 15 y - 5 z = & - 25 \
- 2 x + 6 y + 2 z = & 10
\end{array} \right.$

Accepted Answer

The system of equations is dependent, meaning all equations represent the same plane in three-dimensional space. This is evident because the second and third equations can be derived by multiplying the first equation by 5 and -2, respectively. Therefore, there are infinitely many solutions that satisfy the condition $x - 3y - z = -5$, not just a single point.

Question 16

Without actually solving the problem, choose the correct solution by deciding which choice satisfies the given conditions.
-At the local convenience store, 7 bags of chips and 3 containers of dip cost $\$ 33$. However, 3 bags of chips and 4 containers of dip cost $\$ 25$. What is the cost of one bag of chips and one container of dip?

Accepted Answer

The answer of Without actually solving the problem, choose the...

Question 17

Solve the system.
-$\left\{ \begin{array} { l } 
2 x + 5 y + z = - 18 \
3 x - 3 y - z = 23 \
3 x + y + 4 z = 8
\end{array} \right.$

Accepted Answer

Substituting $x = 3$, $y = -5$, and $z = 1$ into each equation satisfies all three equations, confirming it as the solution.

Question 18

Solve the system.
-

Accepted Answer

The answer of Solve the system.
-
...

Question 19

Solve the system.
-$\left\{ \begin{array} { r r } 
x + y & = - 3 \
- 5 x - 5 y + 5 z & = 35 \
x - z & = - 8
\end{array} \right.$

Accepted Answer

Solving the system of equations, we start by adding the first and third equations to eliminate $z$, leading to $2x = -11$, which gives $x = -\frac{11}{2}$, not matching any of the options. However, if we directly substitute $x = -4$ from option A into the first equation, we get $y = 1$, and substituting $x = -4$ and $y = 1$ into the second equation gives $z = 4$, confirming option A as correct.

Question 20

Write a system of equations in x and y describing the situation. Do not solve the system.
-Bowie has $\$ 11$ less in his pocket than Donna has. If Donna had $\$ 5$ more, she would have 3 times as much money as Bowie.

Accepted Answer

The first equation, $x = y - 11$, represents Bowie having $11 less than Donna. The second equation, $y + 5 = 3x$, represents Donna having 3 times as much money as Bowie if she had $5 more.

Quiz 4: Solving Systems of Linear Equations

Solve the system. - $\left\{ \begin{array} { c } x - y + 2 z = 5 \\5 x + z = 0 \\- x + y - 2 z = - 10\end{array} \right.$

Without actually solving the problem, choose the correct solution by deciding which choice satisfies the given conditions. -Aviva has a total of 70 coins, all of which are either dimes or nickels. The total value of the coins is \$5.95. Find the number of each type of coin.

Solve the system. - $\left\{ \begin{array} { r } y + 5 z = - 8 \\- 4 x + y - 3 z = 8 \\2 x + 3 z = - 7\end{array} \right.$

Solve the system. - $\left\{ \begin{array} { r r } x - y + 5 z & = - 2 \\3 x + z & = 0 \\x + 3 y + z & = 6\end{array} \right.$

Write a system of equations in x and y describing the situation. Do not solve the system. -An order of 2 orders of fries, 3 hamburgers, and 3 drinks costs $\$ 11$ . An order of 4 orders of fries, 2 hamburgers, and 5 drinks costs $\$ 13$ . All drinks are $\$ 1$ .

Write a system of equations in x and y describing the situation. Do not solve the system. -One number is 5 more than another number. If you add 8 to 3 times the first number, the result is 4 times the second number.

Solve. -The sum of two numbers is $- 5$ . Three times the first number equals 4 times the second number. Find the two numbers.

Solve the system. - $\left\{ \begin{aligned}2 x + y - \frac { 3 } { 2 } z & = - 3 \\\frac { 3 } { 2 } x - 3 z & = 0 \\x - 3 y & = - 10\end{aligned} \right.$

Solve the system. - $\left\{ \begin{array} { r } x - y + 3 z = - 5 \\5 x + z = - 1 \\x + 5 y + z = 9\end{array} \right.$

Solve the system. - $\left\{ \begin{array} { l } x - y + z = - 3 \\x + y + z = 7 \\x + y - z = - 1\end{array} \right.$

Solve the system. - $\left\{ \begin{aligned}x + y + z & = - 2 \\x - y + 4 z & = 14 \\4 x + 4 y + 4 z & = - 14\end{aligned} \right.$

Solve the system. - $\left\{ \begin{aligned}x + 2 y + 4 z & = 0 \\5 y + 3 z & = 9 \\z & = - 2\end{aligned} \right.$

Solve the system. - $\left\{ \begin{array} { r } x + y + z = - 10 \\x - y + 4 z = - 13 \\4 x + 4 y + 4 z = - 22\end{array} \right.$

Solve the system. - $\left\{ \begin{array} { r r } x - 3 y - z = & - 5 \\5 x - 15 y - 5 z = & - 25 \\- 2 x + 6 y + 2 z = & 10\end{array} \right.$

Solve the system. - $\left\{ \begin{array} { l } 2 x + 5 y + z = - 18 \\3 x - 3 y - z = 23 \\3 x + y + 4 z = 8\end{array} \right.$

Solve the system. - $Solve the system. - A) ( 0,0 , - 4 ) B) ( 2 , - 4,0 ) C) ( 0 , - 4,0 ) D) \varnothing$

Solve the system. - $\left\{ \begin{array} { r r } x + y & = - 3 \\- 5 x - 5 y + 5 z & = 35 \\x - z & = - 8\end{array} \right.$

Write a system of equations in x and y describing the situation. Do not solve the system. -Bowie has $\$ 11$ less in his pocket than Donna has. If Donna had $\$ 5$ more, she would have 3 times as much money as Bowie.

Quiz 4: Solving Systems of Linear Equations

Solve the system. - {x−y+2z=55x+z=0−x+y−2z=−10\left\{ \begin{array} { c } x - y + 2 z = 5 \\5 x + z = 0 \\- x + y - 2 z = - 10\end{array} \right.⎩⎨⎧​x−y+2z=55x+z=0−x+y−2z=−10​

Without actually solving the problem, choose the correct solution by deciding which choice satisfies the given conditions. -Aviva has a total of 70 coins, all of which are either dimes or nickels. The total value of the coins is \$5.95. Find the number of each type of coin.

Solve the system. - {y+5z=−8−4x+y−3z=82x+3z=−7\left\{ \begin{array} { r } y + 5 z = - 8 \\- 4 x + y - 3 z = 8 \\2 x + 3 z = - 7\end{array} \right.⎩⎨⎧​y+5z=−8−4x+y−3z=82x+3z=−7​

Solve the system. - {x−y+5z=−23x+z=0x+3y+z=6\left\{ \begin{array} { r r } x - y + 5 z & = - 2 \\3 x + z & = 0 \\x + 3 y + z & = 6\end{array} \right.⎩⎨⎧​x−y+5z3x+zx+3y+z​=−2=0=6​

Write a system of equations in x and y describing the situation. Do not solve the system. -An order of 2 orders of fries, 3 hamburgers, and 3 drinks costs $11\$ 11$11 . An order of 4 orders of fries, 2 hamburgers, and 5 drinks costs $13\$ 13$13 . All drinks are $1\$ 1$1 .

Write a system of equations in x and y describing the situation. Do not solve the system. -One number is 5 more than another number. If you add 8 to 3 times the first number, the result is 4 times the second number.

Solve. -The sum of two numbers is −5- 5−5 . Three times the first number equals 4 times the second number. Find the two numbers.

Solve the system. - {2x+y−32z=−332x−3z=0x−3y=−10\left\{ \begin{aligned}2 x + y - \frac { 3 } { 2 } z & = - 3 \\\frac { 3 } { 2 } x - 3 z & = 0 \\x - 3 y & = - 10\end{aligned} \right.⎩⎨⎧​2x+y−23​z23​x−3zx−3y​=−3=0=−10​

Solve the system. - {x−y+3z=−55x+z=−1x+5y+z=9\left\{ \begin{array} { r } x - y + 3 z = - 5 \\5 x + z = - 1 \\x + 5 y + z = 9\end{array} \right.⎩⎨⎧​x−y+3z=−55x+z=−1x+5y+z=9​

Solve the system. - {x−y+z=−3x+y+z=7x+y−z=−1\left\{ \begin{array} { l } x - y + z = - 3 \\x + y + z = 7 \\x + y - z = - 1\end{array} \right.⎩⎨⎧​x−y+z=−3x+y+z=7x+y−z=−1​

Solve the system. - {x+y+z=−2x−y+4z=144x+4y+4z=−14\left\{ \begin{aligned}x + y + z & = - 2 \\x - y + 4 z & = 14 \\4 x + 4 y + 4 z & = - 14\end{aligned} \right.⎩⎨⎧​x+y+zx−y+4z4x+4y+4z​=−2=14=−14​

Solve the system. - {x+2y+4z=05y+3z=9z=−2\left\{ \begin{aligned}x + 2 y + 4 z & = 0 \\5 y + 3 z & = 9 \\z & = - 2\end{aligned} \right.⎩⎨⎧​x+2y+4z5y+3zz​=0=9=−2​

Solve the system. - {x+y+z=−10x−y+4z=−134x+4y+4z=−22\left\{ \begin{array} { r } x + y + z = - 10 \\x - y + 4 z = - 13 \\4 x + 4 y + 4 z = - 22\end{array} \right.⎩⎨⎧​x+y+z=−10x−y+4z=−134x+4y+4z=−22​

Solve the system. - {x−3y−z=−55x−15y−5z=−25−2x+6y+2z=10\left\{ \begin{array} { r r } x - 3 y - z = & - 5 \\5 x - 15 y - 5 z = & - 25 \\- 2 x + 6 y + 2 z = & 10\end{array} \right.⎩⎨⎧​x−3y−z=5x−15y−5z=−2x+6y+2z=​−5−2510​

Solve the system. - {2x+5y+z=−183x−3y−z=233x+y+4z=8\left\{ \begin{array} { l } 2 x + 5 y + z = - 18 \\3 x - 3 y - z = 23 \\3 x + y + 4 z = 8\end{array} \right.⎩⎨⎧​2x+5y+z=−183x−3y−z=233x+y+4z=8​

Solve the system. -

Solve the system. - {x+y=−3−5x−5y+5z=35x−z=−8\left\{ \begin{array} { r r } x + y & = - 3 \\- 5 x - 5 y + 5 z & = 35 \\x - z & = - 8\end{array} \right.⎩⎨⎧​x+y−5x−5y+5zx−z​=−3=35=−8​

Write a system of equations in x and y describing the situation. Do not solve the system. -Bowie has $11\$ 11$11 less in his pocket than Donna has. If Donna had $5\$ 5$5 more, she would have 3 times as much money as Bowie.