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

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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Write an inconsistent system of equations that contains dependent equations.

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