Chapter 6

\(\begin{aligned} \frac{5}{x}-\frac{1}{y}-\frac{2}{z} &=-6 \\ -\frac{1}{x}+\frac{3}{y}-\frac{3}{z} &=-12 \\ \frac{2}{x}-\frac{1}{y}-\frac{1}{z} &=6 \end{aligned}\)

Evaluate each determinant. $$\operatorname{det}\left[\begin{array}{rrrr}4 & 5 & -1 & -1 \\\2 & -3 & 1 & 0 \\\\-5 & 1 & 3 & 9 \\\0 & -2 & 1 & 5\end{array}\right]$$

Solve each system by elimination. $$\begin{array}{r}9 x-5 y=1 \\\\-18 x+10 y=1\end{array}$$

Checking Analytic Skills Graph the solution set of each system of inequalities by hand. Do not use a calculator. $$\begin{array}{r}2 x+y>2 \\\x-3 y<6\end{array}$$

Solve each system by using the matrix inverse method. $$\begin{aligned} &2 x+3 y=-10\\\ &3 x+4 y=-12 \end{aligned}$$

\(\begin{aligned} &x-4 y+2 z=-2\\\ &x+2 y-2 z=-3\\\ &x-y \quad=4 \end{aligned}\)

Evaluate each determinant. $$\operatorname{det}\left[\begin{array}{rrrr}4 & 0 & 0 & 2 \\\\-1 & 0 & 3 & 0 \\\2 & 4 & 0 & 1 \\\0 & 0 & 1 & 2\end{array}\right]$$

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} &2 x-3 y=2\\\ &4 x-6 y=1 \end{aligned}$$

Solve each system by elimination. $$\begin{aligned}&3 x+2 y=5\\\&6 x+4 y=8\end{aligned}$$

Checking Analytic Skills Graph the solution set of each system of inequalities by hand. Do not use a calculator. $$\begin{array}{r}4 x+3 y<12 \\\y+4 x>-4\end{array}$$

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{array}{r} x+2 y=1 \\ 2 x+4 y=3 \end{array}$$

Solve each system by using the matrix inverse method. $$\begin{aligned} &2 x-3 y=10\\\ &2 x+2 y=5 \end{aligned}$$

\(\begin{aligned} 2 x+y+3 z &=4 \\ -3 x-y-4 z &=5 \\ x+y+2 z &=0 \end{aligned}\)

Evaluate each determinant. $$\operatorname{det}\left[\begin{array}{rrrr}-2 & 0 & 4 & 2 \\\3 & 6 & 0 & 4 \\\0 & 0 & 0 & 3 \\\9 & 0 & 2 & -1\end{array}\right]$$

Solve each system by elimination. $$\begin{aligned}&3 x+y=6\\\&6 x+2 y=1\end{aligned}$$

Checking Analytic Skills Graph the solution set of each system of inequalities by hand. Do not use a calculator. $$\begin{array}{r}3 x+5 y \leq 15 \\\x-3 y \geq 9\end{array}$$

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} 6 x-3 y &=1 \\ -12 x+6 y &=-2 \end{aligned}$$

Solve each system by using the matrix inverse method. $$\begin{aligned} &2 x-5 y=10\\\ &2 x-5 y=15 \end{aligned}$$

\(\begin{array}{l} x+y+z=0 \\ x-y-z=3 \\ x+3 y+3 z=5 \end{array}\)

Find matrix \(A\) if $$B=\left[\begin{array}{rrr} 4 & 6 & -5 \\ -6 & 3 & 2 \end{array}\right] \text { and } A+B=\left[\begin{array}{rrr} 6 & 12 & 0 \\ -10 & -4 & 11 \end{array}\right].$$

A triangle with vertices at \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\) and \(\left(x_{3}, y_{3}\right),\) as shown in the figure, has area equal to the absolute value of \(D,\) where $$D=\frac{1}{2} \operatorname{det}\left[\begin{array}{lll}x_{1} & y_{1} & 1 \\\x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1\end{array}\right]$$ Use \(D\) to find the area of each triangle with coordinates as given. $$P(0,0), Q(0,2), R(1,4)$$

Solve each system by elimination. $$\begin{aligned}&3 x+5 y=-2\\\&9 x+15 y=-6\end{aligned}$$

Checking Analytic Skills Graph the solution set of each system of inequalities by hand. Do not use a calculator. $$\begin{array}{r}y \leq x \\\x^{2}+y^{2}<1\end{array}$$

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} x-y &=1 \\ -x+y &=-1 \end{aligned}$$

\(\begin{array}{r} x+3 y+z=6 \\ 3 x+y-z=6 \\ x-y-z=0 \end{array}\)

Find matrix \(B\) if $$A=\left[\begin{array}{rrr} 3 & 6 & 5 \\ -2 & 1 & 4 \end{array}\right] \text { and } A-B=\left[\begin{array}{rrr} 9 & 0 & -5 \\ -4 & 6 & -3 \end{array}\right].$$

A triangle with vertices at \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\) and \(\left(x_{3}, y_{3}\right),\) as shown in the figure, has area equal to the absolute value of \(D,\) where $$D=\frac{1}{2} \operatorname{det}\left[\begin{array}{lll}x_{1} & y_{1} & 1 \\\x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1\end{array}\right]$$ Use \(D\) to find the area of each triangle with coordinates as given. $$P(0,1), Q(2,0), R(1,5)$$

Solve each system by elimination. $$\begin{array}{l}\frac{x}{2}+\frac{y}{3}=8 \\\\\frac{2 x}{3}+\frac{3 y}{2}=17\end{array}$$

Checking Analytic Skills Graph the solution set of each system of inequalities by hand. Do not use a calculator. $$\begin{aligned}4 x-3 y & \leq 12 \\\y & \leq x^{2}\end{aligned}$$

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{array}{l} x+y=-1 \\ y+z=4 \\ x+z=1 \end{array}$$

Solve each system by using the matrix inverse method. $$\begin{aligned} 2 x+4 z &=14 \\ 3 x+y+5 z &=19 \\ -x+y-2 z &=-7 \end{aligned}$$

\(\begin{aligned} 2 x-y+2 z &=6 \\ -x+y+z &=0 \\ -x-3 z &=-6 \end{aligned}\)

The dimensions of matrices \(A\) and \(B\) are given. Find the dimensions of the product \(A B\) and of the product BA if the products are defined. If they are not defined, say so. $$A \text { is } 4 \times 2 ; B \text { is } 2 \times 4.$$

A triangle with vertices at \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\) and \(\left(x_{3}, y_{3}\right),\) as shown in the figure, has area equal to the absolute value of \(D,\) where $$D=\frac{1}{2} \operatorname{det}\left[\begin{array}{lll}x_{1} & y_{1} & 1 \\\x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1\end{array}\right]$$ Use \(D\) to find the area of each triangle with coordinates as given. $$P(2,5), Q(-1,3), R(4,0)$$

Solve each system by elimination. $$\begin{aligned}&\frac{x}{5}+3 y=31\\\&2 x-\frac{y}{5}=8\end{aligned}$$

Checking Analytic Skills Graph the solution set of each system of inequalities by hand. Do not use a calculator. $$\begin{aligned}&y \leq-x^{2}\\\&y \geq x^{2}-6\end{aligned}$$

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} &x-z=-3\\\ &y+z=9\\\ &x+z=7 \end{aligned}$$

Solve each system by using the matrix inverse method. $$\begin{aligned} 3 x+6 y+3 z &=12 \\ 6 x+4 y-2 z &=-4 \\ y-z &=-3 \end{aligned}$$

\(\begin{aligned} x+2 y+z &=0 \\ 3 x+2 y-z &=4 \\ -x+2 y+3 z &=-4 \end{aligned}\)

The dimensions of matrices \(A\) and \(B\) are given. Find the dimensions of the product \(A B\) and of the product BA if the products are defined. If they are not defined, say so. $$A \text { is } 3 \times 1 ; B \text { is } 1 \times 3.$$

A triangle with vertices at \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\) and \(\left(x_{3}, y_{3}\right),\) as shown in the figure, has area equal to the absolute value of \(D,\) where $$D=\frac{1}{2} \operatorname{det}\left[\begin{array}{lll}x_{1} & y_{1} & 1 \\\x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1\end{array}\right]$$ Use \(D\) to find the area of each triangle with coordinates as given. $$P(2,-2), Q(0,0), R(-3,-4)$$

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} x+y & \leq 9 \\ x & \leq-y^{2} \end{aligned}$$

Solve each system by elimination. $$\begin{array}{l}\frac{2 x-1}{3}+\frac{y+2}{4}=4 \\\\\frac{x+3}{2}-\frac{x-y}{3}=3\end{array}$$

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} x+y-z &=6 \\ 2 x-y+z &=-9 \\ x-2 y+3 z &=1 \end{aligned}$$

Solve each system by using the matrix inverse method. $$\begin{aligned} x+3 y+z &=2 \\ x-2 y+3 z &=-3 \\ 2 x-3 y-z &=34 \end{aligned}$$

Feed Requirements Solve the system from Example 4 . $$ \begin{aligned} 25 x+40 y+20 z &=2200 \\ 4 x+2 y+3 z &=280 \\ 3 x+2 y+z &=180 \end{aligned} $$

The dimensions of matrices \(A\) and \(B\) are given. Find the dimensions of the product \(A B\) and of the product BA if the products are defined. If they are not defined, say so. $$A \text { is } 3 \times 5 ; B \text { is } 5 \times 2.$$

A triangle with vertices at \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\) and \(\left(x_{3}, y_{3}\right),\) as shown in the figure, has area equal to the absolute value of \(D,\) where $$D=\frac{1}{2} \operatorname{det}\left[\begin{array}{lll}x_{1} & y_{1} & 1 \\\x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1\end{array}\right]$$ Use \(D\) to find the area of each triangle with coordinates as given. $$P(1,2), Q(4,3), R(3,5)$$

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} &x+2 y \leq 4\\\ &y \geq x^{2}-1 \end{aligned}$$

Solve each system by elimination. $$\begin{aligned}&\frac{x+6}{5}+\frac{2 y-x}{10}=1\\\&\frac{x+2}{4}+\frac{3 y+2}{5}=-3\end{aligned}$$