Section 1
Systems of Equations in Three Variables
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.
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.
Classify each of the following statements as either true or false.Every system of three equations in three unknowns has at least one solution.
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.
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.
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.
Determine whether $(2,-1,-2)$ is a solution of the system$$\begin{aligned}x+y-2 z &=5 \\2 x-y-z &=7 \\-x-2 y-3 z &=6\end{aligned}$$
Determine whether $(-1,-3,2)$ is a solution of the system$$\begin{array}{r}{x-y+z=4} \\{x-2 y-z=3} \\{3 x+2 y-z=1}\end{array}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}x-y-z &=0 \\2 x-3 y+2 z &=7 \\-x+2 y+z &=1\end{aligned}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}x+y-z &=0 \\2 x-y+z &=3 \\-x+5 y-3 z &=2\end{aligned}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}x-y-z &=1 \\2 x+y+2 z &=4 \\x+y+3 z &=5\end{aligned}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}x+y-3 z &=4 \\2 x+3 y+z &=6 \\2 x-y+z &=-14\end{aligned}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}3 x+4 y-3 z &=4 \\5 x-y+2 z &=3 \\x+2 y-z &=-2\end{aligned}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}2 x-3 y+z &=5 \\x+3 y+8 z &=22 \\3 x-y+2 z &=12\end{aligned}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}x+y+z &=0 \\2 x+3 y+2 z &=-3 \\-x-2 y-z &=1\end{aligned}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}3 a-2 b+7 c &=13 \\a+8 b-6 c &=-47 ,\\7 a-9 b-9 c &=-3\end{aligned}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}2 x-3 y-z &=-9 ,\\2 x+5 y+z &=1 ,\\x-y+z &=3\end{aligned}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}4 x+y+z &=17 ,\\x-3 y+2 z &=-8, \\5 x-2 y+3 z &=5\end{aligned}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}a+b+c &=5, \\2 a+3 b-c &=2 ,\\2 a+3 b-2 c &=4\end{aligned}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}u-v+6 w &=8 ,\\3 u-v+6 w &=14 ,\\-u-2 v-3 w &=7\end{aligned}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{array}{r}{-2 x+8 y+2 z=4} ,\\{x+6 y+3 z=4} ,\\{3 x-2 y+z=0}\end{array}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}x-y+z &=4 ,\\5 x+2 y-3 z &=2 ,\\4 x+3 y-4 z &=-2\end{aligned}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{array}{ll}{2 u-4 v-w=8} ,\\{3 u+2 v+w=6} ,\\{5 u-2 v+3 w=2}\end{array}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}&4 a+b+c=3,\\&2 a-b+c=6,\\&2 a+2 b-c=-9\end{aligned}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}r+\frac{3}{2} s+6 t &=2, \\2 r-3 s+3 t &=0.5, \\r+s+t &=1\end{aligned}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}5 x+3 y+\frac{1}{2} z &=\frac{7}{2} ,\\0.5 x-0.9 y-0.2 z &=0.3 ,\\3 x-2.4 y+0.4 z &=-1\end{aligned}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}&4 a+9 b=8,\\&8 a+6 c=-1,\\&6 b+6 c=-1\end{aligned}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}&3 u+2 w=11,\\&v-7 w=4,\\&u-6 v=1\end{aligned}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}x+y+z &=57 ,\\-2 x+y \quad \quad &=3 ,\\x- \quad \quad \quad& z=6\end{aligned}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}x \quad+y+z &=105 ,\\10 y-z &=11 ,\\2 x-3 y \quad \quad&=7\end{aligned}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}a \quad \quad -3 c=6 ,\\b+2 c =2 ,\\7 a-3 b-5 c =14\end{aligned}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{array}{l}{2 a-3 b=2} ,\\{7 a+4 c=\frac{3}{4}} ,\\{2 c-3 b=1}\end{array}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}&x+y+z=83,\\&y=2 x+3,\\&z=40+x\end{aligned}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}&l+m=7,\\&3 m+2 n=9,\\&4 l+n=5\end{aligned}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}x \quad \quad+z &=0 ,\\x+y+2 z &=3 ,\\y+ \quad z &=2\end{aligned}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}x+y \quad \quad =0 ,\\x \quad \quad+z=1 ,\\2 x+y+z =2\end{aligned}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}x+y+z &=1 ,\\-x+2 y+z &=2 ,\\2 x-y \quad \quad &=-1\end{aligned}$$
Solve each system. If a system’s equations are dependent or if there is no solution, state this.$$\begin{aligned}\quad \quad y +z &=1 ,\\x+y+z &=1 ,\\x+2 y+2 z &=2\end{aligned}$$
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?
Describe a method for writing an inconsistent system of three equations in three variables.
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.
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.
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.
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 $100 .$
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.
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.
Is it possible for a system of three linear equations to have exactly two ordered triples in its solution set? Why or why not?
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?
Solve.$$\begin{aligned}&\frac{x+2}{3}-\frac{y+4}{2}+\frac{z+1}{6}=0,\\&\frac{x-4}{3}+\frac{y+1}{4}+\frac{z-2}{2}=-1,\\&\frac{x+1}{2}+\frac{y}{2}+\frac{z-1}{4}=\frac{3}{4}\end{aligned} $$
Solve.$$\begin{aligned}&w+x+y+z=2,\\&w+2 x+2 y+4 z=1,\\&w-x+y+z=6,\\&w-3 x-y+z=2\end{aligned}$$
Solve.$$\begin{aligned}w+x-y+z &=0 ,\\w-2 x-2 y-z &=-5 ,\\w-3 x-y+z &=4 ,\\2 w-x-y+3 z &=7\end{aligned}$$
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 $z$.$$\begin{aligned}&\frac{2}{x}-\frac{1}{y}-\frac{3}{z}=-1\\&\frac{2}{x}-\frac{1}{y}+\frac{1}{z}=-9\\&\frac{1}{x}+\frac{2}{y}-\frac{4}{z}=17\end{aligned}$$
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 $z$.$$\begin{aligned}&\frac{2}{x}+\frac{2}{y}-\frac{3}{z}=3,\\&\frac{1}{x}-\frac{2}{y}-\frac{3}{z}=9,\\&\frac{7}{x}-\frac{2}{y}+\frac{9}{z}=-39\end{aligned}$$
Determine $k$ so that each system is dependent.$$\begin{aligned}x-3 y+2 z &=1 ,\\2 x+y-z &=3 ,\\9 x-6 y+3 z &=k\end{aligned}$$
Determine $k$ so that each system is dependent.$$\begin{aligned}5 x-6 y+k z &=-5 ,\\x+3 y-2 z &=2 ,\\2 x-y+4 z &=-1\end{aligned}$$
In each case, three solutions of an equation in $x, y,$ and $z$ are given. Find the equation.$$\begin{aligned}&A x+B y+C z=12 ;\\&\left(1, \frac{3}{4}, 3\right),\left(\frac{4}{3}, 1,2\right), \text { and }(2,1,1)\end{aligned}$$
In each case, three solutions of an equation in $x, y,$ and $z$ are given. Find the equation.$$\begin{aligned}&z=b-m x-n y ;\\&(1,1,2),(3,2,-6), \text { and }\left(\frac{3}{2}, 1,1\right)\end{aligned}$$
Write an inconsistent system of equations that contains dependent equations.