Elementary and Intermediate Algebra

Marvin L. Bittinger, David J. Ellenbogen,Barbara L. Johnson

Chapter 9

More on Systems

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.

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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.

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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.

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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.

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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.

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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.

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Problem 7

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

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Problem 8

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

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Problem 9

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

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Problem 10

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

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Problem 11

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

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Problem 12

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

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Problem 13

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

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Problem 14

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

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Problem 15

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

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Problem 16

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

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Problem 17

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

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Problem 18

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

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Problem 19

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

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Problem 20

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

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Problem 21

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

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Problem 22

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

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Problem 23

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

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Problem 24

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

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Problem 25

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

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Problem 26

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

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Problem 27

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

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Problem 28

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

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Problem 29

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

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Problem 30

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

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Problem 31

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

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Problem 32

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

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Problem 33

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

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Problem 34

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

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Problem 35

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

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Problem 36

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

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Problem 37

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

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Problem 38

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

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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?

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Problem 40

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

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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.

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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.

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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.

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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 $100 .$

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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.

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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.

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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?

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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?

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Problem 49

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

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Problem 50

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

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Problem 51

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

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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 $z$.
$$\begin{aligned}
&\frac{2}{x}-\frac{1}{y}-\frac{3}{z}=-1\\
&\frac{2}{x}-\frac{1}{y}+\frac{1}{z}=-9\\
&\frac{1}{x}+\frac{2}{y}-\frac{4}{z}=17
\end{aligned}$$

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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 $z$.
$$\begin{aligned}
&\frac{2}{x}+\frac{2}{y}-\frac{3}{z}=3,\\
&\frac{1}{x}-\frac{2}{y}-\frac{3}{z}=9,\\
&\frac{7}{x}-\frac{2}{y}+\frac{9}{z}=-39
\end{aligned}$$

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Problem 54

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

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Problem 55

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

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Problem 56

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

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Problem 57

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

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Problem 58

Write an inconsistent system of equations that contains dependent equations.

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