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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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

Karly W.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

Write an inconsistent system of equations that contains dependent equations.

Karly W.

Numerade Educator