Chapter 7

Solve each system by using the matrix inverse method. $$\begin{aligned} x+3 y &=-12 \\ 2 x-y &=11 \end{aligned}$$

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

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} &4 x+y=5\\\ &2 x+y=3 \end{aligned}$$

Find the partial fraction decomposition for each rational expression. $$\frac{3 x^{6}+3 x^{4}+3 x}{x^{4}+x^{2}}$$

Solve each system by elimination. $$\begin{aligned}&6 x+7 y=-2\\\&7 x-6 y=26\end{aligned}$$

Which one of the following is a description of the graph of the inequality \((x-5)^{2}+(y-2)^{2}<4 ?\) A. The region inside a circle with center \((-5,-2)\) and radius 2 B. The region inside a circle with center \((5,2)\) and radius 2 C. The region inside a circle with center \((-5,-2)\) and radius 4 D. The region outside a circle with center \((5,2)\) and radius 4

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

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 analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} 2 x+3 y+4 z &=3 \\ 6 x+3 y+8 z &=6 \\ 6 y-4 z &=1 \end{aligned}$$

Solve each system by elimination. $$\begin{aligned}x-y &=9 \\\\-2 x+2 y &=-18\end{aligned}$$

Which one of the given inequalities satisfies the follow. ing description: the region outside a circle centered at the origin, with \(x\) -intercepts \((4,0)\) and \((-4,0) ?\). A. \(x^{2}+y^{2}>4\) B. \((x-4)^{2}+y^{2}>16\) C. \(x^{2}+y^{2}<16\) D. \(x^{2}+y^{2}>16\)

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

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

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

Determine whether each partial fraction decomposition is correct by graphing the left side and the right side of the equation on the same coordinate axes and observing whether the graphs coincide. $$\frac{1}{(x-1)(x+2)}=\frac{1}{x-1}-\frac{1}{x+2}$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} 10 x+2 y-3 z &=0 \\ 5 x+4 y+6 z &=-1 \\ 6 y+3 z &=2 \end{aligned}$$

Solve each system by elimination. $$\begin{aligned}x+y &=4 \\\3 x+3 y &=12\end{aligned}$$

Write an inequality that satisfies the description.Inside the circle with radius 1 and center \((0,0)\)?

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

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 cqual 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 coondinates as given. (GRAPH CAN'T COPY) $$P(0,0), Q(0,2), R(1,4)$$

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

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

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{array}{r} \frac{1}{x}+\frac{1}{y}-\frac{1}{z}=\frac{1}{4} \\ \frac{2}{x}-\frac{1}{y}+\frac{3}{z}=\frac{9}{4} \\ -\frac{1}{x}-\frac{2}{y}+\frac{4}{z}=1 \end{array}$$

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

Write an inequality that satisfies the description.Outside the circle with radius 3 and center \((0,0)\).

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 cqual 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 coondinates as given. (GRAPH CAN'T COPY) $$P(0,1), Q(2,0), R(1,5)$$

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

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

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} &\frac{3}{x}+\frac{2}{y}-\frac{1}{z}=\frac{11}{6}\\\ &\frac{1}{x}-\frac{1}{y}+\frac{3}{z}=-\frac{11}{12}\\\ &\frac{2}{x}+\frac{1}{y}+\frac{1}{z}=\frac{7}{12} \end{aligned}$$

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

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

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 cqual 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 coondinates as given. (GRAPH CAN'T COPY) $$P(2,5), Q(-1,3), R(4,0)$$

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

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. $$A \text { is } 4 \times 2 ; B \text { is } 2 \times 4$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} &\frac{2}{x}-\frac{2}{y}+\frac{1}{z}=-1\\\ &\frac{4}{x}+\frac{1}{y}-\frac{2}{z}=-9\\\ &\frac{1}{x}+\frac{1}{y}-\frac{3}{z}=-9 \end{aligned}$$

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

Write an inequality that satisfies the description.Below the parabola with vertex \((0,1)\) and \(x\) -intercepts \((-1,0)\) and \((1,0)\).

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

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 cqual 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 coondinates as given. (GRAPH CAN'T COPY) $$P(2,-2), Q(0,0), R(-3,-4)$$

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

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. $$A \text { is } 3 \times 1 ; B \text { is } 1 \times 3$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} \frac{5}{x}-\frac{1}{y}-\frac{2}{z} &=-6 \\ -\frac{1}{x}+\frac{3}{y}-\frac{3}{z} &=-12 \\ \frac{2}{x}-\frac{1}{y}-\frac{1}{z} &=6 \end{aligned}$$

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

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

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 cqual 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 coondinates as given. (GRAPH CAN'T COPY) $$P(1,2), Q(4,3), R(3,5)$$

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

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. $$A \text { is } 3 \times 5 ; B \text { is } 5 \times 2$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} &\frac{3}{x}+\frac{2}{y}-\frac{1}{z}=\frac{11}{6}\\\ &\frac{1}{x}-\frac{1}{y}+\frac{3}{z}=-\frac{11}{12}\\\ &\frac{2}{x}+\frac{1}{y}+\frac{1}{z}=\frac{7}{12} \end{aligned}$$

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