Deck 6: Systems of Linear Equations and Matrices

Question 1

Determine whether the given ordered set of numbers is a solution of the system of equations.

-(-6,-2)
$\mathrm{x}+\mathrm{y}=-8$
$\mathrm{x}-\mathrm{y}=-4$

Answer 1

True

Question 2

Determine whether the given ordered set of numbers is a solution of the system of equations.

- $(-6,-3)$
$x+y=3$
$\mathrm{x}-\mathrm{y}=9$

Answer 2

False

Question 3

Determine whether the given ordered set of numbers is a solution of the system of equations.

- $(-1,2)$
$4 x+y=-2$
$2 x+4 y=6$

Answer 3

True

Question 4

Determine whether the given ordered set of numbers is a solution of the system of equations.

- $(4,3)$
$3 \mathrm{x}+\mathrm{y}=9$
$2 x+3 y=-1$

Question 5

Determine whether the given ordered set of numbers is a solution of the system of equations.

- $(-2,-1,3)$
$4 x-4 y+z=-1$
$5 \mathrm{x}+5 \mathrm{z}=5$
$x+4 y-2 z=-12$

Question 6

Determine whether the given ordered set of numbers is a solution of the system of equations.

- $(3,-4,2.5)$
$4 \mathrm{x}-3 \mathrm{y}+\mathrm{z}=24.5$
$5 \mathrm{x}+2 \mathrm{z}=20$
$0.5 \mathrm{x}+2 \mathrm{y}-2 \mathrm{z}=-9.5$

Question 7

Solve the system of two equations in two variables.

- $x-3 y=10$
$-7 x-4 y=30$

A) $(2,-3)$
B) $(-2,-4)$
C) No solution
D) $(-3,-3)$

Question 8

Solve the system of two equations in two variables.

- $x+9 y=-18$
$8 x+10 y=-20$

A) No solution
B) $(0,-2)$
C) $(2,0)$
D) $(1,-3)$

Question 9

Solve the system of two equations in two variables.

- $5 x+9 y=43$
$-2 x-7 y=-24$

A) $(5,3)$
B) $(4,3)$
C) $(5,2)$
D) No solution

Question 10

Solve the system of two equations in two variables.

- $7 x+5 y=-5$
$-3 x-2 y=2$

A) $(-1,0)$
B) $(0,-1)$
C) No solution
D) $(0,0)$

Question 11

Solve the system of two equations in two variables.

- $-7 x-7 y=-56$
$-2 x+5 y=-16$

A) No solution
B) $(7,1)$
C) $(8,1)$
D) $(8,0)$

Question 12

Solve the system of two equations in two variables.

- $6 x-98=8 y$
$-3 x+5 y=-56$

A) $(7,-7)$
B) $(7,-6)$
C) $(6,-6)$
D) No solution

Question 13

Solve the system of two equations in two variables.

- $4 x-2 y=3$
$20 x-10 y=12$

A) $(0,-1.5)$
B) $(1,0)$
C) No solution
D) $(1,0.5)$

Question 14

Solve the system of two equations in two variables.

- $x+y=-6$
$x-y=19$

A) $\left(6,-\frac{25}{2}\right)$
B) $\left(\frac{13}{2}, \frac{25}{2}\right)$
C) $\left(\frac{13}{2},-\frac{25}{2}\right)$
D) $\left(6, \frac{13}{2}\right)$

Question 15

Solve the system of two equations in two variables.

- $4 x+3 y=2$
$24 x+18 y=12$

A) No solution
B) $\left(\frac{1}{2}, 0\right)$
C) $\left(-\frac{1}{4}, 1\right)$
D) $\left(-\frac{3}{4} y+\frac{1}{2}, y\right)$ for any real number $y$

Question 16

Multiply both sides of each equation by a common denominator to eliminate the fractions. Then solve the system.

- $\frac{1}{5} x+\frac{1}{5} y=-1$
$x-y=-9$

A) $(-8,3)$
B) No solution
C) $(-7,2)$
D) $(7,3)$

Question 17

Multiply both sides of each equation by a common denominator to eliminate the fractions. Then solve the system.

- $\frac{1}{3} x+\frac{1}{3} y=0$
$\frac{1}{3} \mathrm{x}-\frac{1}{3} \mathrm{y}=\frac{4}{3}$

A) $(1,-1)$
B) $(2,-2)$
C) $(-2,-1)$
D) No solution

Question 18

Multiply both sides of each equation by a common denominator to eliminate the fractions. Then solve the system.

- $\frac{3 x}{8}-\frac{3 y}{5}=\frac{33}{80}$
$\frac{4 x}{7}+\frac{4 y}{5}=\frac{37}{35}$

A) $\left(\frac{1}{4}, \frac{1}{2}\right)$
B) $\left(\frac{3}{4}, \frac{1}{2}\right)$
C) $\left(\frac{3}{2}, \frac{3}{4}\right)$
D) $\left(\frac{3}{2}, \frac{1}{4}\right)$

Question 19

Multiply both sides of each equation by a common denominator to eliminate the fractions. Then solve the system.

- $\frac{3 x}{2}-\frac{y}{3}=-18$
$\frac{3 x}{4}+\frac{2 y}{9}=-9$

A) $(0,-12)$
B) $(-12,0)$
C) $(0,12)$
D) $(12,0)$

Question 20

Multiply both sides of each equation by a common denominator to eliminate the fractions. Then solve the system.

- $\frac{7 x}{3}+\frac{5 y}{4}=4$
$\frac{5 x}{6}-2 y=21$

A) $(6,8)$
B) $(-6,8)$
C) $(6,-8)$
D) $(-6,-8)$

Question 21

Multiply both sides of each equation by a common denominator to eliminate the fractions. Then solve the system.

- $\frac{5 x}{2}-\frac{5 y}{4}=-\frac{5}{2}$
$\frac{8 x}{9}=\frac{4}{9}$

A) $\left(\frac{1}{2},-3\right)$
B) $\left(-\frac{1}{2},-3\right)$
C) $\left(-\frac{1}{2}, 3\right)$
D) $\left(\frac{1}{2}, 3\right)$

Question 22

Multiply both sides of each equation by a common denominator to eliminate the fractions. Then solve the system.

- $3 x-\frac{5 y}{7}=10$
$\frac{2 x}{3}-\frac{9 y}{7}=\frac{19}{5}$

A) $\left(3, \frac{7}{5}\right)$
B) $\left(3, \frac{7}{9}\right)$
C) $\left(3,-\frac{7}{5}\right)$
D) $\left(3,-\frac{7}{9}\right)$

Question 23

Solve the problem by writing and solving a suitable system of equations.

-Best Rentals charges a daily fee plus a mileage fee for renting its cars. Barney was charged $\$ 159$ for 3 days and 300 miles, while Mary was charged $\$ 289$ for 5 days and 600 miles. What does Best Rental charge per day and per mile?

A) $\$ 28$ per day and 25 cents per mile
B) $\$ 24$ per day and 29 cents per mile
C) $\$ 30$ per day and 25 cents per mile
D) $\$ 29$ per day and 24 cents per mile

Question 24

Solve the problem by writing and solving a suitable system of equations.

-A shopkeeper orders 18 pounds of cashews and peanuts. If the amount of cashews he orders is 14 pounds less than the amount of peanuts, how many pounds of peanuts did he order?

A) 4 pounds
B) 9 pounds
C) 2 pounds
D) 16 pounds

Question 25

Solve the problem by writing and solving a suitable system of equations.

-Carole's car averages 13.0 miles per gallon in city driving and 21.0 miles per gallon in highway driving. If she drove a total of 443.0 miles on 23 gallons of gas, how many of the gallons were used for city driving?

A) 23 gallons
B) 7 gallons
C) 18 gallons
D) 5 gallons

Question 26

Solve the problem by writing and solving a suitable system of equations.

-If 40 pounds of tomatoes and 20 pounds of bananas cost $\$ 26$ and 10 pounds of tomatoes and 30 pounds of bananas cost $\$ 14$ , what is the price per pound of tomatoes and bananas

A) tomatoes: $\$ 0.60$ per pound; bananas: $\$ 0.10$ per pound
B) tomatoes: $\$ 0.40$ per pound; bananas: $\$ 0.50$ per pound
C) tomatoes: $\$ 0.60$ per pound; bananas: $\$ 0.30$ per pound
D) tomatoes: $\$ 0.50$ per pound; bananas: $\$ 0.30$ per pound

Question 27

Obtain an equivalent system by performing the stated elementary operation on the system.

-Interchange equations 1 and 3.
$5 x+5 y+z=7$
$5 x-4 y-z=-33$
$3 x+3 z=1$

A) $3 x \quad+3 z=1$
$5 x-4 y-z=-33$
$5 x+5 y+z=7$
B) $3 x \quad+3 z=1$
$5 x+5 y+z=7$
$5 x-4 y-z=-33$
C) $x+5 y+5 z=7$
$5 x-4 y-z=-33$
$3 x+3 z=1$
D) $5 x-4 y-z=-33$
$5 x+5 y+z=7$
$3 x+3 z=1$

Question 28

Obtain an equivalent system by performing the stated elementary operation on the system.

-Multiply the second equation by -1 .
$x-5 y+z=5$
$3 x-3 y-z=-18$
$5 x+y+4 z=-12$
$x-3 y+z=-7$

A) x - 5y + z = 5
-3x + 3y + z = 18
5x + y + 4z = -12
X - 3y + z = -7
B) x - 5y + z = 5
-3x - 3y - z = -18
5x + y + 4z = -12
X - 3y + z = -7
C) x - 5y + z = 5
-3x + 3y + z = -18
5x + y + 4z = -12
X - 3y + z = -7
D) -x + 5y - z = -5
3x - 3y - z = -18
5x + y + 4z = -12
X - 3y + z = -7

Question 29

Obtain an equivalent system by performing the stated elementary operation on the system.

-Multiply the third equation by $1 / 8$ .
$7 x-y+8 z+w=-10$
$8 \mathrm{x} \quad-\mathrm{z}-2 \mathrm{w}=7$
$8 x-16 y+7 z-w=-24$
$\mathrm{x}+3 \mathrm{y}-8 \mathrm{z}=1$

A) $7 x-y+8 z+w=-10$
$8 x \quad-z-2 w=7$
$\mathrm{x}-16 \mathrm{y}+7 \mathrm{z}-\mathrm{w}=-24$
$\mathrm{x}+3 \mathrm{y}-8 \mathrm{z}=1$
B) $7 x-y+8 z+w=-10$
$8 x-z-2 w=7$
$x-2 y+\frac{7}{8} z-\frac{1}{8} w=-24$
$x+3 y-8 z \quad=1$
C) $7 x-y+8 z+w=-10$
$8 x-z-2 w=7$
$x-2 y+\frac{7}{8} z-\frac{1}{8} w=-3$
$\mathrm{x}+3 \mathrm{y}-8 \mathrm{z}=1$
D) $7 x-y+8 z+w=-10$
$8 x-z-2 w=7$
$x-2 y+\frac{7}{8} z+\frac{1}{8} w=-24$
$\mathrm{x}+3 \mathrm{y}-8 \mathrm{z} \quad=1$

Question 30

Obtain an equivalent system by performing the stated elementary operation on the system.

-Replace the third equation by the sum of itself and -1 times the second equation.
$x-2 y-7 z=17$
$-6 x+4 y+5 z=-9$
$8 x+7 y-z=-4$

A) $x-2 y-7 z=17$
$14 x+3 y-6 z=5$
$8 x+7 y-z=-4$
B) $x-2 y-7 z=17$
$-6 x+4 y+5 z=-9$
$14 x+3 y-6 z=5$
C) $x-2 y-7 z=17$
$-6 x+4 y+5 z=-9$
$2 x+11 y+4 z=-13$
D) $x-2 y-7 z=17$
$-6 x+4 y+5 z=-9$
$-14 x-3 y+6 z=5$

Question 31

Obtain an equivalent system by performing the stated elementary operation on the system.

-Replace the fourth equation by the sum of itself and 3 times the second equation
X-2 y+5 z-6 w=4 \\
4 y-z+4 w=-5 \\
3 y-4 z+2 w=-3 \\
2 y-2 z-3 w=8

A)x-2 y+5 z-6 w & =4 \\
4 y-z+4 w & =-5 \\
3 y-4 z+2 w & =-3 \\
14 y-5 z+9 w & =-7
B)x-2 y+5 z-6 w= & 4 \\
12 y-3 z+12 w= & -15 \\
3 y-4 z+2 w= & -3 \\
2 y-2 z-3 w= & 8
C)x-2 y+5 z-6 w=4 \\
4 y-z+4 w=-5 \\
3 y-4 z+2 w=-3 \\
12 y+3 z+9 w=-7
D)x-2 y+5 z-6 w=4 \\
4 y-z+4 w=-5 \\
3 y-4 z+2 w=-3 \\
-10 y+5 z-15 w=23

Question 32

Solve the system by back substitution.

- $x+4 y+4 z=11$
$3 y+5 z=17$
$2 z=8$

A) No solution
B) $(-6,-1,4)$
C) $(-1,4,-1)$
D) $(-1,-1,4)$

Question 33

Solve the system by back substitution.

-6 x+2 y-3 z-5 w & =12
Y-2 z-5 w & =-11
5 z-2 w & =-24
4 w & =8

A) $\left(-\frac{4}{3}, 9,-4,2\right)$
B) $\left(\frac{14}{3},-9,-4,2\right)$
C) $(2,-9,-4,2)$
D) $(6,-13,-4,2)$

Question 34

Write an augmented matrix for the system of equations.

- $9 x+4 y=5$
$8 x-2 y=10$

A) $\left[\begin{array}{rr|r}9 & 4 & 5 \\ 8 & -2 & 10\end{array}\right]$
B) $\left[\begin{array}{rr|r}5 & 4 & 9 \\ 10 & 8 & -2\end{array}\right]$
C) $\left[\begin{array}{rr|r}9 & 8 & 5 \\ 4 & -2 & 10\end{array}\right]$
D) $\left[\begin{array}{rr|r}9 & 4 & 10 \\ -2 & 8 & 5\end{array}\right]$

Question 35

Write an augmented matrix for the system of equations.

- $2 x+4 y=22$
$4 y=4$

A) $\left[\begin{array}{rr|r}22 & 4 & 2 \\ 4 & 0 & 4\end{array}\right]$
B) $\left[\begin{array}{rr|r}2 & 4 & 22 \\ 0 & 4 & 4\end{array}\right]$
C) $\left[\begin{array}{rr|r}2 & 4 & 22 \\ 4 & 4 & 0\end{array}\right]$
D) $\left[\begin{array}{ll|l}4 & 0 & 4 \\ 2 & 4 & 4\end{array}\right]$

Question 36

Write an augmented matrix for the system of equations.

- $2 x+9 y+6 z=32$
$4 x+6 y+4 z=24$
$3 x+2 y+8 z=63$

A)
$\left[\begin{array}{lll|l}2 & 4 & 3 & 32 \\9 & 6 & 2 & 24 \\6 & 4 & 8 & 63\end{array}\right]$
B)
$\left[\begin{array}{lll|l}2 & 9 & 6 & 32 \\ 4 & 6 & 4 & 24 \\ 3 & 2 & 8 & 63\end{array}\right]$
C)
$\left[\begin{array}{lll}2 & 9 & 6 \\ 4 & 6 & 4 \\ 3 & 2 & 8\end{array}\right]$
D)
$\left[\begin{array}{lll|l}32 & 6 & 9 & 2 \\ 24 & 4 & 6 & 4 \\ 63 & 8 & 2 & 3\end{array}\right]$

Question 37

Write an augmented matrix for the system of equations.

- $9 x+9 z=54$
$-2 y+8 z=2$
$7 x+7 y+3 z=83$

A)
$\left[\begin{array}{rrr|r}9 & 0 & 9 & 54 \\ 0 & -2 & 8 & 2 \\ 7 & 7 & 3 & 83\end{array}\right]$
B) $\left[\begin{array}{rrr}9 & 0 & 9 \\ 0 & -2 & 8 \\ 7 & 7 & 3\end{array}\right]$
C)
$\left[\begin{array}{rrr|r}9 & 0 & 7 & 54 \\ 0 & -2 & 7 & 2 \\ 9 & 8 & 3 & 83\end{array}\right]$
D)
$\left[\begin{array}{rrr|r}9 & 9 & 0 & 54 \\ -2 & 8 & 0 & 2 \\ 7 & 7 & 3 & 83\end{array}\right]$

Question 38

Write the system of equations associated with the augmented matrix. Do not solve.

- $\left[\begin{array}{rr|r}1 & 0 & -10 \\ 0 & 1 & 9\end{array}\right]$

A) $x=0$
$\mathrm{y}=0$
B) $x=-10$
$y=9$
C) $x=10$
$y=-9$
D) $x=1$
$\mathrm{y}=1$

Question 39

Write the system of equations associated with the augmented matrix. Do not solve.

- $\left[\begin{array}{rrr|r}1 & 0 & 0 & 6 \\ 0 & 1 & 0 & -7 \\ 0 & 0 & 1 & 9\end{array}\right]$

A) $x=6$
$y=-7$
$\mathrm{z}=9$
B) $x=0$
$y=-1$
$\mathrm{z}=15$
C) $x=-3$
$y=-16$
$\mathrm{z}=0$
D) $x=-6$
$\mathrm{y}=7$
$\mathrm{z}=-9$

Question 40

Perform the row operations on the matrix and write the resulting matrix.

-Replace $R_{2}$ by $R_{1}+(-1) R_{2}$
$\left[\begin{array}{rr|r}1 & -3 & 4 \\ 2 & 3 & 1\end{array}\right]$

A)
$\left[\begin{array}{rr|r}1 & -2 & 2 \\ 3 & 1 & 3\end{array}\right]$
B)
$\left[\begin{array}{rr|r}1 & -2 & 2 \\ 1 & 5 & -1\end{array}\right]$
C)
$\left[\begin{array}{rr|r}1 & -3 & 4 \\ -1 & -6 & 3\end{array}\right]$
D)
$\left[\begin{array}{rr|r}1 & -2 & 2 \\ 2 & 3 & 1\end{array}\right]$

Question 41

Perform the row operations on the matrix and write the resulting matrix.

-Replace $R_{2}$ by $R_{1}+R_{2}$
$\left[\begin{array}{rr|r}1 & 0 & 2 \\ -1 & 1 & 3\end{array}\right]$

A) $\left[\begin{array}{ll|l}1 & 0 & 2 \\ 0 & 0 & 5\end{array}\right]$
B) $\left[\begin{array}{ll|l}1 & 0 & 2 \\ 0 & 1 & 5\end{array}\right]$
C) $\left[\begin{array}{rr|r}0 & 1 & 5 \\ -1 & 1 & 3\end{array}\right]$
D) $\left[\begin{array}{rr|r}1 & 0 & 2 \\ -1 & 1 & 3\end{array}\right]$

Question 42

Perform the row operations on the matrix and write the resulting matrix.

-Replace $R_{2}$ by $\frac{1}{2} R_{1}+\frac{1}{2} R_{2}$
$\left[\begin{array}{rr|r}2 & 0 & 6 \\ -2 & 2 & 12\end{array}\right]$

A) $\left[\begin{array}{ll|l}2 & 0 & 6 \\ 0 & 0 & 9\end{array}\right]$
B) $\left[\begin{array}{ll|l}2 & 0 & 6 \\ 0 & 2 & 18\end{array}\right]$
C) $\left[\begin{array}{ll|l}2 & 0 & 6 \\ 0 & 1 & 9\end{array}\right]$
D) $\left[\begin{array}{rr|r}2 & 0 & 6 \\ -1 & 1 & 6\end{array}\right]$

Question 43

Perform the row operations on the matrix and write the resulting matrix.

-Replace $R_{3}$ by $\frac{1}{2} R_{1}+R_{2}$
$\left[\begin{array}{rrr|r}2 & 2 & 4 & 6 \\ 1 & -1 & -2 & 5 \\ 0 & 1 & 0 & 1\end{array}\right]$

A) $\left[\begin{array}{rrr|r}2 & 2 & 4 & 6 \\1 & -1 & -2 & 5 \\2 & 0 & 0 & 8\end{array}\right]$
B) $\left[\begin{array}{lll|r}1 & 1 & 2 & 3 \\ 2 & 0 & 0 & 8 \\ 0 & 1 & 0 & 1\end{array}\right]$
C) $\left[\begin{array}{lll|r}2 & 2 & 4 & 6 \\ 0 & 1 & 0 & 1 \\ 3 & 1 & 2 & 11\end{array}\right]$
D) $\left[\begin{array}{lll|l}1 & 1 & 2 & 3 \\ 0 & 1 & 0 & 0 \\ 2 & 0 & 0 & 8\end{array}\right]$

Question 44

Perform the row operations on the matrix and write the resulting matrix.

-Replace $R_{2}$ by $\frac{1}{3} R_{1}+\frac{1}{2} R_{2}$
$\left[\begin{array}{rr|r}3 & 0 & 9 \\ -2 & 4 & 8\end{array}\right]$

A) $\left[\begin{array}{ll|l}3 & 0 & 9 \\ 0 & 2 & 7\end{array}\right]$
B) $\left[\begin{array}{ll|l}3 & 0 & 9 \\ 0 & 0 & 7\end{array}\right]$
C) $\left[\begin{array}{rr|r}3 & 0 & 9 \\ -1 & 2 & 4\end{array}\right]$
D) $\left[\begin{array}{rr|r}3 & 0 & 9 \\ 1 & 4 & 17\end{array}\right]$

Question 45

The reduced row echelon form of the augmented matrix of a system of equations is given. Find the solutions of thesystem.

-45 $\left[\begin{array}{llll|c}1 & 0 & 0 & 0 & 12 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -8 \\ 0 & 0 & 0 & 1 & 11 / 2\end{array}\right]$

A) $\left(12, w,-8, \frac{11}{2}\right)$ for any real number $w$
B) $\left(12,0,-8, \frac{11}{2}\right)$
C) $\left(12,-8, \frac{11}{2}, 0\right)$
D) No solution

Question 46

The reduced row echelon form of the augmented matrix of a system of equations is given. Find the solutions of thesystem.

- $\left[\begin{array}{rrrrr|r}1 & 0 & 0 & 0 & 0 & 2 \\ 0 & 1 & 0 & 0 & 0 & 8 \\ 0 & 0 & 1 & 0 & 0 & 4 \\ 0 & 0 & 0 & 1 & 0 & -4 \\ 0 & 0 & 0 & 0 & 0 & 1\end{array}\right]$

A) $(2,8,4,-4,1)$
B) $(2,8,4,-4, w)$ for any real number $\mathrm{w}$
C) $(2,8,4,-4)$
D) No solution

Question 47

The reduced row echelon form of the augmented matrix of a system of equations is given. Find the solutions of thesystem.

- $\left[\begin{array}{rrrr|r}1 & 0 & 0 & 3 & 12 \\ 0 & 1 & 0 & 0 & -8 \\ 0 & 0 & 1 & -2 & 6 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]$

A) $(12+3 w,-8,6-2 w$ , w) for any real number $w$
B) No solution
C) $(12,-8,6,0)$
D) $(12-3 w,-8,6+2 w$ , w) for any real number $w$

Question 48

The reduced row echelon form of the augmented matrix of a system of equations is given. Find the solutions of thesystem.

- $\left[\begin{array}{rrrr|r}1 & 0 & 0 & 0 & -2 \\ 0 & 1 & 0 & 0 & 19 \\ 0 & 0 & 1 & 0 & 6 \\ 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]$

A) $(-2,19,6,2, w)$ for any real number $\mathrm{w}$
B) $(-2,19,6,2)$
C) $(-2,19,6,2,0)$
D) No solution

Question 49

Perform row operations on the augmented matrix as far as necessary to determine whether the system is independent,dependent, or inconsistent.

- $x+y+z=-1$
$x-y+3 z=-5$
$3 x+y+z=-3$

A)I nconsistent
B) Independent
C) Dependent

Question 50

Perform row operations on the augmented matrix as far as necessary to determine whether the system is independent,dependent, or inconsistent.

- $x-y+4 z=4$
$4 \mathrm{x}+\mathrm{z}=2$
$x+4 y+z=18$

A) Dependent
B) Inconsistent
C) Independent

Question 51

Perform row operations on the augmented matrix as far as necessary to determine whether the system is independent,dependent, or inconsistent.

- $x-y+z=0$
$x+y+z=-10$
$x+y-z=-2$

A) Dependent
B) Independent
C) Inconsistent

Question 52

Perform row operations on the augmented matrix as far as necessary to determine whether the system is independent,dependent, or inconsistent.

- $x+5 y+5 z=-9$
$2 y+2 z=-2$
$\mathrm{z}=4$

A) Independent
B) Inconsistent
C) Dependent

Question 53

Perform row operations on the augmented matrix as far as necessary to determine whether the system is independent,dependent, or inconsistent.

- $x+y+z=1$
$x-y+5 z=-1$
$4 x+4 y+4 z=10$

A) Dependent
B) Independent
C) Inconsistent

Question 54

Perform row operations on the augmented matrix as far as necessary to determine whether the system is independent,dependent, or inconsistent.

- $-x-y-z=-6$
$x+y+z=0$
$\mathrm{x}+\mathrm{y}-\mathrm{z}=4$

A) Dependent
B) Independent
C) Inconsistent

Question 55

Perform row operations on the augmented matrix as far as necessary to determine whether the system is independent,dependent, or inconsistent.

- $x-y+z=-6$
$2 x+y+z=0$
$-x+y-z=15$

A) Dependent
B) Inconsistent
C) Independent

Question 56

Perform row operations on the augmented matrix as far as necessary to determine whether the system is independent,dependent, or inconsistent.

- $x+y-2 z=8$
$3 x+z=-6$
$2 x-y+3 z=-14$

A) Independent
B) Inconsistent
C) Dependent

Question 57

Perform row operations on the augmented matrix as far as necessary to determine whether the system is independent,dependent, or inconsistent.

- $x+y+z=7$
$x-y+2 z=7$
$2 x+3 z=14$

A) Independent
B) Inconsistent
C) Dependent

Question 58

Solve the system of equations. If the system is dependent, express solutions in terms of the parameter $\mathrm{z}$ .

- $x+y+z=-3$
$x-y+2 z=-1$
$3 x+y+z=-1$

A) $(-3,1,-1)$
B) No solution
C) $(1,-2,-2)$
D) $(-3,-1,1)$

Question 59

Solve the system of equations. If the system is dependent, express solutions in terms of the parameter $\mathrm{z}$ .

- $x+y+z=2$
$x-y+5 z=12$
$5 \mathrm{x}+\mathrm{y}+\mathrm{z}=-6$

A) No solution
B) $(3,-2,1)$
C) $(3,1,-2)$
D) $(-2,1,3)$

Question 60

Solve the system of equations. If the system is dependent, express solutions in terms of the parameter $\mathrm{z}$ .

- $x-y+4 z=-22$
$4 \mathrm{x}+\mathrm{z}=-5$
$x+3 y+z=1$

A) $(-5,0,2)$
B) No solution
C) $(-5,2,0)$
D) $(0,2,-5)$

Question 61

Solve the system of equations. If the system is dependent, express solutions in terms of the parameter $\mathrm{z}$ .

- $x-y+z=1$
$x+y+z=-5$
$\mathrm{x}+\mathrm{y}-\mathrm{z}=-11$

A) $(3,-5,-3)$
B) $(-5,3,-3)$
C) $(-5,-3,3)$
D) No solution

Question 62

Solve the system of equations. If the system is dependent, express solutions in terms of the parameter $\mathrm{z}$ .

- $x+4 y+3 z=-6$
$3 y+2 z=-8$
$z=2$

A) No solution
B) $(4,2,-4)$
C) $(2,-4,4)$
D) $(4,-4,2)$

Question 63

Solve the system of equations. If the system is dependent, express solutions in terms of the parameter $\mathrm{z}$ .

- $2 x+y-z=2$
$x-3 y+2 z=1$
$7 \mathrm{x}-7 \mathrm{y}+4 \mathrm{z}=7$

A) $(2,5,7)$
B) $(1,0,0)$
C) No solution
D) $\left(\frac{7+z}{7}, \frac{5}{7} z, z\right)$ for any real number $z$

Question 64

Solve the system of equations. If the system is dependent, express solutions in terms of the parameter $\mathrm{z}$ .

- $-2 x+4 y+7 z=8$
$7 \mathrm{x}+\mathrm{y}-6 \mathrm{z}=15$

A) $\left(\frac{26}{15}+\frac{31}{30} z, 15+7 x-6 z, z\right)$
B) $\left(\frac{26}{15}+\frac{31}{30} z, 15-7 x+6 z, z\right)$
C) $\left(\frac{26}{15}+\frac{31}{30} z, \frac{43}{15}-\frac{37}{30} z, z\right)$
D) $(-52-31 z,-86-52 z, z)$

Question 65

Solve the system of equations. If the system is dependent, express solutions in terms of the parameter $\mathrm{z}$ .

- $2 x+y-2 z=12$
$4 \mathrm{x}-4 \mathrm{y}+6 \mathrm{z}=14$

A) $\left(\frac{31}{6}+\frac{1}{6} z, \frac{5}{3}+\frac{5}{3} z, z\right)$
B) $\left(\frac{31}{6}+\frac{1}{6} z, 12-2 x+2 z, z\right)$
C) $\left(\frac{31}{6}+\frac{1}{6} z, 12+2 x-2 z, z\right)$
D) $(62+2 z, 20+62 z, z)$

Question 66

Use the Gauss-Jordan method to solve the system of equations.

- $x+y+z=-12$
$x-y+3 z=-8$
$4 x+y+z=-24$

A) $(-4,-5,-3)$
B) No solution
C) $(-3,-4,-5)$
D) $(-3,-5,-4)$

Question 67

Use the Gauss-Jordan method to solve the system of equations.

- $x-y+5 z=-17$
$5 \mathrm{x}+\mathrm{z}=-3$
$x+4 y+z=5$

A) No solution
B) $(0,2,-3)$
C) $(-3,0,2)$
D) $(-3,2,0)$

Question 68

Use the Gauss-Jordan method to solve the system of equations.

- $x-y+z=-7$
$x+y+z=1$
$\mathrm{x}+\mathrm{y}-\mathrm{z}=11$

A) $(2,-5,4)$
B) $(-5,2,4)$
C) No solution
D) $(2,4,-5)$

Question 69

Use the Gauss-Jordan method to solve the system of equations.

- $x+4 y+3 z=11$
$3 y+4 z=19$
$\mathrm{z}=4$

A) $(4,1,-5)$
B) $(-5,1,4)$
C) No solution
D) $(-5,4,1)$

Question 70

Use the Gauss-Jordan method to solve the system of equations.

- $9 x+6 y-z=60$
$x+5 y+3 z=55$
$-7 \mathrm{x}+\mathrm{y}+\mathrm{z}=5$

A) No solution
B) $(1,9,3)$
C) $(1,3,9)$
D) $(-1,9,2)$

Question 71

Use the Gauss-Jordan method to solve the system of equations.

- $-3 x-y-9 z=-75$
$3 x+5 y-2 z=44$
$-8 \mathrm{x}-6 \mathrm{y}+\mathrm{z}=-95$

A) No solution
B) $(8,6,5)$
C) $(-8,6,16)$
D) $(8,5,6)$

Question 72

Use the Gauss-Jordan method to solve the system of equations.

- $7 x-y-7 z=4$
$-7 x+3 z=-39$
$7 y+z=29$

A) $(-9,3,18)$
B) No solution
C) $(9,3,8)$
D) $(9,8,3)$

Question 73

Use the Gauss-Jordan method to solve the system of equations.

- $x+y+z=9$
$2 x-3 y+4 z=7$
$x-4 y+3 z=-2$

A) $\left(\frac{7 z+34}{5}, \frac{2 z-11}{5}, z\right)$
B) $\left(\frac{-7 z+34}{5}, \frac{2 z+11}{5}, z\right)$
C) $\left(\frac{-7 z+34}{5}, \frac{2 z-11}{5}, z\right)$
D) $\left(\frac{7 z+34}{5}, \frac{2 z+11}{5}, z\right)$

Question 74

Use the Gauss-Jordan method to solve the system of equations.

- $3 x+y+z=5$
$4 \mathrm{x}+5 \mathrm{y}-\mathrm{z}=-8$
$10 \mathrm{x}+7 \mathrm{y}+\mathrm{z}=2$

A) $\left(\frac{6 z+33}{11}, \frac{7 z+44}{11}, z\right)$
B) $\left(\frac{6 z+33}{11}, \frac{7 z-44}{11}, z\right)$
C) $\left(\frac{-6 z+33}{11}, \frac{7 z-44}{11}, z\right)$
D) $\left(\frac{-6 z+33}{11}, \frac{-7 z-44}{11}, z\right)$

Question 75

Use the Gauss-Jordan method to solve the system of equations.

- $x+y+z=7$
$x-y+2 z=7$
$2 x+3 z=15$

A) No solution
B) $(2,1,4)$
C) $(1,2,4)$
D) $(4,2,1)$

Question 76

Use the Gauss-Jordan method to solve the system of equations.

- $x-y+3 z=-8$
$x+5 y+z=40$
$5 x+y+13 z=10$

A) $(8,0,8)$
B) No solution
C) $(0,8,0)$
D) $(8,8,0)$

Question 77

Solve the problem by writing and solving a suitable system of equations.

-Alan invests a total of $\$ 18,000$ in three different ways. He invests one part in a mutual fund which in the first year has a return of $11 \%$ . He invests the second part in a government bond at $7 \%$ per year. The third part he puts in the bank at $5 \%$ per year. He invests twice as much in the mutual fund as in the bank. The first year Alan's investments bring a total return of $\$ 1500$ . How much did he invest in each way?

A)m utual fund: $\$ 7400$ ; bond: $\$ 6900$ : bank: $\$ 3700$
B)m utual fund: $\$ 8000$ ; bond: $\$ 6000$ : bank: $\$ 4000$
C)m utual fund: $\$ 8000$ ; bond: $\$ 7000$ : bank: $\$ 4000$
D)m utual fund: $\$ 8600$ ; bond: $\$ 5100$ : bank: $\$ 4300$

Question 78

Solve the problem by writing and solving a suitable system of equations.

-Julia is preparing a meal by combining three ingredients. One unit of each ingredient provides the following quantities (in grams) of carbohydrates, fat, and protein.

Ideally the meal should contain 25 grams of protein, 35 grams of carbohydrates, and 13 grams of fat. How many units of each ingredient should Julia use?

A) 2 grams of ingredient $A, 4$ grams of ingredient $B, 3$ grams of ingredient $C$
B) 3 grams of ingredient $A$ , 4 grams of ingredient $B$ , 2 grams of ingredient $C$
C) 3 grams of ingredient $A$ , 2 grams of ingredient $B, 4$ grams of ingredient $C$
D) 4 grams of ingredient $A, 3$ grams of ingredient $B, 2$ grams of ingredient $C$

Question 79

Solve the problem by writing and solving a suitable system of equations.

-A company produces three models of video cassette player, models $X, Y$ , and Z. Each model $X$ machine requires 3.2 hours of electronics work, 2.8 hours of assembly time, and 4.4 hours of quality assurance time. Each model $\mathrm{Y}$ machine requires 5.2 hours of electronics work, 4.4 hours of assembly time, and 5.2 hours of quality assurance time. Each model $\mathrm{Z}$ machine requires 5.2 hours of electronics work, 3.2 hours of assembly time, and 3.8 hours of quality assurance time. There are 440 hours available each week for electronics, 346 hours for assembly, and 453 hours for quality assurance. How many of each model should be produced each week if all available time must be used?

A) 38 model $X$ , 37 model Y, 25 model $Z$
B) 41 model X, 35 model Y, 24 model Z
C) 40 model $X$ , 30 model $Y, 30$ model $Z$
D) 40 model $X$ , 35 model Y, 25 model Z

Question 80

Solve the problem by writing and solving a suitable system of equations.

-Barges from ports $X$ and $Y$ went to cities $A$ and $B$ . $X$ sent 30 barges and $Y$ sent 8 . City A needs 21 barges and B needs 17. Shipping costs $\$ 220$ from $X$ to $A, \$ 300$ from $X$ to $B, \$ 400$ from $Y$ to $A$ , and $\$ 180$ from $Y$ to B. $\$ 8760$ was spent. How many barges went where?

A)
B)
C)
D)