# Elementary and Intermediate Algebra

## Educators

### Problem 1

Classify each of the following statements as either true or false.
$3 x+5 y+4 z=7$ is a linear equation in three variables.

Karly W.

### Problem 2

Classify each of the following statements as either true or false.
It is not difficult to solve a system of three equations in three unknowns by graphing.

Karly W.

### Problem 3

Classify each of the following statements as either true or false.
Every system of three equations in three unknowns has at least one solution.

Karly W.

### Problem 4

Classify each of the following statements as either true or false.
If, when we are solving a system of three equations, a false equation results from adding a multiple of one equation to another, the system is inconsistent.

Karly W.

### Problem 5

Classify each of the following statements as either true or false.
If, when we are solving a system of three equations, an identity results from adding a multiple of one equation to another, the equations are dependent.

Karly W.

### Problem 6

Classify each of the following statements as either true or false.
Whenever a system of three equations contains dependent equations, there is an infinite number of solutions.

Karly W.

### Problem 7

Determine whether $(2,-1,-2)$ is a solution of the system
\begin{aligned} x+y-2 z &=5 \\ 2 x-y-z &=7 \\ -x-2 y-3 z &=6 \end{aligned}

Karly W.

### Problem 8

Determine whether $(-1,-3,2)$ is a solution of the system
$$\begin{array}{r} {x-y+z=4} \\ {x-2 y-z=3} \\ {3 x+2 y-z=1} \end{array}$$

Karly W.

### Problem 9

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} x-y-z &=0 \\ 2 x-3 y+2 z &=7 \\ -x+2 y+z &=1 \end{aligned}

Karly W.

### Problem 10

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} x+y-z &=0 \\ 2 x-y+z &=3 \\ -x+5 y-3 z &=2 \end{aligned}

Karly W.

### Problem 11

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} x-y-z &=1 \\ 2 x+y+2 z &=4 \\ x+y+3 z &=5 \end{aligned}

Karly W.

### Problem 12

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} x+y-3 z &=4 \\ 2 x+3 y+z &=6 \\ 2 x-y+z &=-14 \end{aligned}

Karly W.

### Problem 13

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} 3 x+4 y-3 z &=4 \\ 5 x-y+2 z &=3 \\ x+2 y-z &=-2 \end{aligned}

Karly W.

### Problem 14

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} 2 x-3 y+z &=5 \\ x+3 y+8 z &=22 \\ 3 x-y+2 z &=12 \end{aligned}

Karly W.

### Problem 15

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} x+y+z &=0 \\ 2 x+3 y+2 z &=-3 \\ -x-2 y-z &=1 \end{aligned}

Karly W.

### Problem 16

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} 3 a-2 b+7 c &=13 \\ a+8 b-6 c &=-47 ,\\ 7 a-9 b-9 c &=-3 \end{aligned}

Karly W.

### Problem 17

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} 2 x-3 y-z &=-9 ,\\ 2 x+5 y+z &=1 ,\\ x-y+z &=3 \end{aligned}

Karly W.

### Problem 18

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} 4 x+y+z &=17 ,\\ x-3 y+2 z &=-8, \\ 5 x-2 y+3 z &=5 \end{aligned}

Karly W.

### Problem 19

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} a+b+c &=5, \\ 2 a+3 b-c &=2 ,\\ 2 a+3 b-2 c &=4 \end{aligned}

Karly W.

### Problem 20

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} u-v+6 w &=8 ,\\ 3 u-v+6 w &=14 ,\\ -u-2 v-3 w &=7 \end{aligned}

Karly W.

### Problem 21

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
$$\begin{array}{r} {-2 x+8 y+2 z=4} ,\\ {x+6 y+3 z=4} ,\\ {3 x-2 y+z=0} \end{array}$$

Karly W.

### Problem 22

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} x-y+z &=4 ,\\ 5 x+2 y-3 z &=2 ,\\ 4 x+3 y-4 z &=-2 \end{aligned}

Karly W.

### Problem 23

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
$$\begin{array}{ll} {2 u-4 v-w=8} ,\\ {3 u+2 v+w=6} ,\\ {5 u-2 v+3 w=2} \end{array}$$

Karly W.

### Problem 24

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} &4 a+b+c=3,\\ &2 a-b+c=6,\\ &2 a+2 b-c=-9 \end{aligned}

Karly W.

### Problem 25

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} r+\frac{3}{2} s+6 t &=2, \\ 2 r-3 s+3 t &=0.5, \\ r+s+t &=1 \end{aligned}

Karly W.

### Problem 26

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} 5 x+3 y+\frac{1}{2} z &=\frac{7}{2} ,\\ 0.5 x-0.9 y-0.2 z &=0.3 ,\\ 3 x-2.4 y+0.4 z &=-1 \end{aligned}

Karly W.

### Problem 27

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} &4 a+9 b=8,\\ &8 a+6 c=-1,\\ &6 b+6 c=-1 \end{aligned}

Karly W.

### Problem 28

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} &3 u+2 w=11,\\ &v-7 w=4,\\ &u-6 v=1 \end{aligned}

Karly W.

### Problem 29

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} x+y+z &=57 ,\\ -2 x+y \quad \quad &=3 ,\\ x- \quad \quad \quad& z=6 \end{aligned}

Karly W.

### Problem 30

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} x \quad+y+z &=105 ,\\ 10 y-z &=11 ,\\ 2 x-3 y \quad \quad&=7 \end{aligned}

Karly W.

### Problem 31

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} a \quad \quad -3 c=6 ,\\ b+2 c =2 ,\\ 7 a-3 b-5 c =14 \end{aligned}

Karly W.

### Problem 32

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
$$\begin{array}{l} {2 a-3 b=2} ,\\ {7 a+4 c=\frac{3}{4}} ,\\ {2 c-3 b=1} \end{array}$$

Karly W.

### Problem 33

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} &x+y+z=83,\\ &y=2 x+3,\\ &z=40+x \end{aligned}

Karly W.

### Problem 34

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} &l+m=7,\\ &3 m+2 n=9,\\ &4 l+n=5 \end{aligned}

Karly W.

### Problem 35

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} x \quad \quad+z &=0 ,\\ x+y+2 z &=3 ,\\ y+ \quad z &=2 \end{aligned}

Karly W.

### Problem 36

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} x+y \quad \quad =0 ,\\ x \quad \quad+z=1 ,\\ 2 x+y+z =2 \end{aligned}

Karly W.

### Problem 37

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} x+y+z &=1 ,\\ -x+2 y+z &=2 ,\\ 2 x-y \quad \quad &=-1 \end{aligned}

Karly W.

### Problem 38

Solve each system. If a system’s equations are dependent or if there is no solution, state this.
\begin{aligned} \quad \quad y +z &=1 ,\\ x+y+z &=1 ,\\ x+2 y+2 z &=2 \end{aligned}

Karly W.

### Problem 39

Rondel always begins solving systems of three equations in three variables by using the first two equations to eliminate $x$. Is this a good approach? Why or why not?

Karly W.

### Problem 40

Describe a method for writing an inconsistent system of three equations in three variables.

Karly W.

### Problem 41

To prepare for Section 9.2, review translating sentences to equations (Section 1.1).
Translate each sentence to an equation.$[1.1]$
One number is half another.

Karly W.

### Problem 42

To prepare for Section 9.2, review translating sentences to equations (Section 1.1).
Translate each sentence to an equation.$[1.1]$
The difference of two numbers is twice the first number.

Karly W.

### Problem 43

To prepare for Section 9.2, review translating sentences to equations (Section 1.1).
Translate each sentence to an equation.$[1.1]$
The sum of three consecutive numbers is 100.

Karly W.

### Problem 44

To prepare for Section 9.2, review translating sentences to equations (Section 1.1).
Translate each sentence to an equation.$[1.1]$
The sum of three numbers is $100 .$

Karly W.

### Problem 45

To prepare for Section 9.2, review translating sentences to equations (Section 1.1).
Translate each sentence to an equation.$[1.1]$
The product of two numbers is five times a third number.

Karly W.

### Problem 46

To prepare for Section 9.2, review translating sentences to equations (Section 1.1).
Translate each sentence to an equation.$[1.1]$
The product of two numbers is twice their sum.

Karly W.

### Problem 47

Is it possible for a system of three linear equations to have exactly two ordered triples in its solution set? Why or why not?

Karly W.

### Problem 48

Kadi and Ahmed both correctly solve the system
\begin{aligned} x+2 y-z &=1 ,\\ -x-2 y+z &=3 ,\\ 2 x+4 y-2 z &=2. \end{aligned}
Kadi states “the equations are dependent” while Ahmed states “there is no solution.” How did each person reach the conclusion?

Karly W.

### Problem 49

Solve.
\begin{aligned} &\frac{x+2}{3}-\frac{y+4}{2}+\frac{z+1}{6}=0,\\ &\frac{x-4}{3}+\frac{y+1}{4}+\frac{z-2}{2}=-1,\\ &\frac{x+1}{2}+\frac{y}{2}+\frac{z-1}{4}=\frac{3}{4} \end{aligned}

Karly W.

### Problem 50

Solve.
\begin{aligned} &w+x+y+z=2,\\ &w+2 x+2 y+4 z=1,\\ &w-x+y+z=6,\\ &w-3 x-y+z=2 \end{aligned}

Karly W.

### Problem 51

Solve.
\begin{aligned} w+x-y+z &=0 ,\\ w-2 x-2 y-z &=-5 ,\\ w-3 x-y+z &=4 ,\\ 2 w-x-y+3 z &=7 \end{aligned}

Karly W.

### Problem 52

For Exercises 52 and $53,$ let u represent $1 / x,$ v represent $1 / y,$ and $w$ represent $1 / z .$ Solve for $u, v,$ and $w,$ and then solve for $x, y,$ and $z$.
\begin{aligned} &\frac{2}{x}-\frac{1}{y}-\frac{3}{z}=-1\\ &\frac{2}{x}-\frac{1}{y}+\frac{1}{z}=-9\\ &\frac{1}{x}+\frac{2}{y}-\frac{4}{z}=17 \end{aligned}

Karly W.

### Problem 53

For Exercises 52 and $53,$ let u represent $1 / x,$ v represent $1 / y,$ and $w$ represent $1 / z .$ Solve for $u, v,$ and $w,$ and then solve for $x, y,$ and $z$.
\begin{aligned} &\frac{2}{x}+\frac{2}{y}-\frac{3}{z}=3,\\ &\frac{1}{x}-\frac{2}{y}-\frac{3}{z}=9,\\ &\frac{7}{x}-\frac{2}{y}+\frac{9}{z}=-39 \end{aligned}

Karly W.

### Problem 54

Determine $k$ so that each system is dependent.
\begin{aligned} x-3 y+2 z &=1 ,\\ 2 x+y-z &=3 ,\\ 9 x-6 y+3 z &=k \end{aligned}

Karly W.

### Problem 55

Determine $k$ so that each system is dependent.
\begin{aligned} 5 x-6 y+k z &=-5 ,\\ x+3 y-2 z &=2 ,\\ 2 x-y+4 z &=-1 \end{aligned}

Karly W.

### Problem 56

In each case, three solutions of an equation in $x, y,$ and $z$ are given. Find the equation.
\begin{aligned} &A x+B y+C z=12 ;\\ &\left(1, \frac{3}{4}, 3\right),\left(\frac{4}{3}, 1,2\right), \text { and }(2,1,1) \end{aligned}

Karly W.

### Problem 57

In each case, three solutions of an equation in $x, y,$ and $z$ are given. Find the equation.
\begin{aligned} &z=b-m x-n y ;\\ &(1,1,2),(3,2,-6), \text { and }\left(\frac{3}{2}, 1,1\right) \end{aligned}

Karly W.