Chapter 6

Find each matrix product if possible. $$\left[\begin{array}{rr} -4 & 0 \\ 1 & 3 \end{array}\right]\left[\begin{array}{rr} -2 & 4 \\ 0 & 1 \end{array}\right]$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{aligned}&4 x-y=0\\\&2 x+3 y=14\end{aligned}$$

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

Solve each system graphically. Check your solutions. Do not use a calculator. $$\begin{aligned}&x^{2}+y^{2}=5\\\&x+y=3\end{aligned}$$

Given a square matrix \(A^{-1}\), find matrix \(A\). $$A^{-1}=\left[\begin{array}{rr} 5 & -9 \\ -1 & 2 \end{array}\right]$$

Solve each system. Write solutions in terms of \(z\) if necessary. $$\begin{aligned} x+2 y-z &=0 \\ 3 x-y+z &=6 \\ -2 x-4 y+2 z &=0 \end{aligned}$$

Find the equation of the parabola with vertical axis that passes through the data points shown or specified. Check your answer. $$(-1,4),(1,2),(3,8)$$

Find each matrix product if possible. $$\left[\begin{array}{rrr} 2 & 2 & -1 \\ 3 & 0 & 1 \end{array}\right]\left[\begin{array}{rr} 0 & 2 \\ -1 & 4 \\ 0 & 2 \end{array}\right]$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{aligned}&3 x+2 y=4\\\&6 x+4 y=8\end{aligned}$$

Solve each system graphically. Check your solutions. Do not use a calculator. $$\begin{aligned}&x-y=3\\\&x^{2}+y^{2}=9\end{aligned}$$

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

Given a square matrix \(A^{-1}\), find matrix \(A\). $$A^{-1}=\left[\begin{array}{rr} \frac{3}{20} & \frac{1}{4} \\ -\frac{1}{20} & \frac{1}{4} \end{array}\right]$$

Solve each system. Write solutions in terms of \(z\) if necessary. $$\begin{aligned} 3 x+5 y-z &=0 \\ 4 x-y+2 z &=1 \\ -6 x-10 y+2 z &=0 \end{aligned}$$

Find the equation of the parabola with vertical axis that passes through the data points shown or specified. Check your answer. $$(-2,2),(0,2),(2,-6)$$

Find each matrix product if possible. $$\left[\begin{array}{rrr} -9 & 2 & 1 \\ 3 & 0 & 0 \end{array}\right]\left[\begin{array}{r} 2 \\ -1 \\ 4 \end{array}\right]$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{array}{r}1.5 x+3 y=5 \\\2 x+4 y=3\end{array}$$

Graph the solution set of each system of inequalities by hand. Concept Check \(\quad\) Which one of the choices that follow is a description of the solution set of the following system? $$ x^{2}+y^{2} < 36 $$ \(y < x\) A. All points outside the circle \(x^{2}+y^{2}=36\) and above the line \(y=x\) B. All points outside the circle \(x^{2}+y^{2}=36\) and below the line \(y=x\) C. All points inside the circle \(x^{2}+y^{2}=36\) and above the line \(y=x\) D. All points inside the circle \(x^{2}+y^{2}=36\) and below the line \(y=x\)

Solve each system graphically. Check your solutions. Do not use a calculator. $$\begin{aligned}&x^{2}-y=0\\\&x^{2}+y^{2}=2\end{aligned}$$

Given a square matrix \(A^{-1}\), find matrix \(A\). $$A^{-1}=\left[\begin{array}{rrr} \frac{2}{3} & -\frac{1}{3} & 0 \\ \frac{1}{3} & -\frac{5}{3} & 1 \\ \frac{1}{3} & \frac{1}{3} & 0 \end{array}\right]$$

Solve each system. Write solutions in terms of \(z\) if necessary. $$\begin{aligned} x-2 y+z &=5 \\ -2 x+4 y-2 z &=2 \\ 2 x+y-z &=2 \end{aligned}$$

Find the equation of the parabola with vertical axis that passes through the data points shown or specified. Check your answer. $$(0,1),(1,0),(2,-5)$$

Find each matrix product if possible. $$\left[\begin{array}{rrr} -2 & -3 & -4 \\ 2 & -1 & 0 \\ 4 & -2 & 3 \end{array}\right]\left[\begin{array}{rrr} 0 & 1 & 4 \\ 1 & 2 & -1 \\ 3 & 2 & -2 \end{array}\right]$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{array}{r}2 x-3 y=-5 \\\x+5 y=17\end{array}$$

Concept Check Fill in each blank with the appropriate response. The graph of the system $$ y > x^{2}+2 $$ $$ \begin{array}{r} x^{2}+y^{2}<16 \\ y<7 \end{array} $$ consists of all points \(\frac{\text { the parabola given by }}{\text { (above/below) }}\) \(y=x^{2}+2, \frac{\underline{\phantom{xx}}}{\text { (inside/outside) }}\) the circle \(x^{2}+y^{2}=16,\) and (above/below) the line \(y =7\)

Solve each system graphically. Check your solutions. Do not use a calculator. $$\begin{aligned}&x-y^{2}=1\\\&x^{2}+y^{2}=5\end{aligned}$$

Given a square matrix \(A^{-1}\), find matrix \(A\). $$A^{-1}=\left[\begin{array}{lll} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right]$$

Solve each system. Write solutions in terms of \(z\) if necessary. $$\begin{aligned} 3 x+6 y-3 z &=12 \\ -x-2 y+z &=16 \\ x+y-2 z &=20 \end{aligned}$$

The values in the table are from a quadratic function \(f(x)=a x^{2}+b x+c .\) Find \(a, b,\) and \(c\). $$\begin{array}{c|c|c|c|c|c}x & -2 & -1 & 0 & 1 & 2 \\\\\hline f(x) & 2.9 & 1.26 & 0.56 & 0.8 & 1.98 \end{array}$$

Find each matrix product if possible. $$\left[\begin{array}{rrr} -1 & 2 & 0 \\ 0 & 3 & 2 \\ 0 & 1 & 4 \end{array}\right]\left[\begin{array}{rrr} 2 & -1 & 2 \\ 0 & 2 & 1 \\ 3 & 0 & -1 \end{array}\right]$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{array}{l}x+9 y=-15 \\\3 x+2 y=5\end{array}$$

Solve each system graphically. Check your solutions. Do not use a calculator. $$\begin{aligned}&x^{2}+y^{2}=4\\\&x+y=2\end{aligned}$$

Solve each system of four equations in four variables. Express the solutions in the form \((x, y, z, w)\) $$\begin{aligned} x+3 y-2 z-w &=9 \\ 4 x+y+z+2 w &=2 \\ -3 x-y+z-w &=-5 \\\ x-y-3 z-2 w &=2 \end{aligned}$$

Given three noncollinear points, there is one and only one circle that passes through them. Knowing that the equation of a circle may be written in the form. $$x^{2}+y^{2}+a x+b y+c=0$$ $$\begin{aligned} &\text {find the equation of the circle passing through the points}\\\ &\text { shown or specified.} \end{aligned}$$ (GRAPH CAN'T COPY)

Find each matrix product if possible. $$\left[\begin{array}{lll} -2 & 4 & 1 \end{array}\right]\left[\begin{array}{rrr} 3 & -2 & 4 \\ 2 & 1 & 0 \\ 0 & -1 & 4 \end{array}\right]$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{aligned}&4 x-y+3 z=-3\\\&3 x+y+z=0\\\&2 x-y+4 z=0\end{aligned}$$

Given a square matrix \(A^{-1}\), find matrix \(A\). Let \(A=\left[\begin{array}{lll}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array}\right],\) where \(a, b,\) and \(c\) are nonzero real numbers. Find \(A^{-1}\).

Solve each system graphically. Check your solutions. Do not use a calculator. $$\begin{aligned}&x^{2}-y=0\\\&x+y^{2}=0\end{aligned}$$

Given a square matrix \(A^{-1}\), find matrix \(A\). Let \(A=\left[\begin{array}{rrr}1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & -1\end{array}\right] .\) Show that \(A^{3}=I_{3},\) and use this result to find the inverse of \(A\).

Solve each system of four equations in four variables. Express the solutions in the form \((x, y, z, w)\) $$\begin{aligned} 3 x+2 y-w &=0 \\ 2 x+z+2 w &=5 \\ x+2 y-z &=-2 \\ 2 x-y+z+w &=2 \end{aligned}$$

Find each matrix product if possible. $$\left[\begin{array}{lll} 0 & 3 & -4 \end{array}\right]\left[\begin{array}{rrr} -2 & 6 & 3 \\ 0 & 4 & 2 \\ -1 & 1 & 4 \end{array}\right]$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{array}{l}5 x+2 y+z=15 \\\2 x-y+z=9 \\\4 x+3 y+2 z=13\end{array}$$

Solve each nonlinear system of equations analytically. $$\begin{aligned}&y=-x^{2}+2\\\&x-y=0\end{aligned}$$

Find each matrix product if possible. $$\left[\begin{array}{ll} p & q \\ r & s \end{array}\right]\left[\begin{array}{ll} a & c \\ b & d \end{array}\right]$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{aligned}2 x-y+4 z &=-2 \\\3 x+2 y-z &=-3 \\\x+4 y+2 z &=17\end{aligned}$$

Solve each nonlinear system of equations analytically. $$\begin{aligned}&y=(x-1)^{2}\\\&x-3 y=-1\end{aligned}$$

Solve each application. Paid Vacation for Employees The average number \(y\) of paid days off for full- time workers at medium-to-large companies after \(x\) years is listed in the table. $$\begin{array}{|l|l|l|l} \hline x \text { (years) } & 1 & 15 & 30 \\ \hline y \text { (days) } & 9.4 & 18.8 & 21.9 \end{array}$$ A. Determine the coefficients for \(f(x)=a x^{2}+b x+c\) so that \(f\) models these data. B. Graph function \(f\) with the data in the viewing window \([-4,32]\) by \([8,23]\) C. Estimate the number of paid days off after 3 years of experience. Compare it with the actual value of 11.2 days

Find each matrix product if possible. $$\left[\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{aligned}x+y+z &=4 \\\2 x-y+3 z &=4 \\\4 x+2 y-z &=-15\end{aligned}$$

Use the shading capabilities of your graphing calculator to graph each inequality or system of inequalities. $$3 x+2 y \geq 6$$

Solve each nonlinear system of equations analytically. $$\begin{aligned}&x^{2}+y^{2}=5\\\&x-y=1\end{aligned}$$