Chapter 6

Draw a sketch of the two graphs described with the indicated number of points of intersection. (There may be more than one way to do this.) A line and a circle; one point.

Graph the solution set of each system of inequalities by hand. $$\begin{array}{l} -2 < x < 3 \\ -1 \leq y \leq 5 \\ 2 x+y < 6 \end{array}$$

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} 3 x-13 y &=17 \\ -23 x+15 y &=2 \end{aligned}$$

A student won \(\$ 100,000\) in the Louisiana state lottery. He invested part of the money in real estate with an annual return of \(5 \%\) and another part in a money market account at \(0.5 \%\) interest. He invested the rest, which amounted to \(\$ 20,000\) less than the sum of the other two parts, in certificates of deposit that pay \(1.75 \%\) If the total annual interest on the money was \(\$ 3250,\) how much was invested at each rate?

If possible, find \(A B\) and \(B A.\) $$A=\left[\begin{array}{rrr} 2 & 1 & -1 \\ 0 & 2 & 1 \\ 3 & 2 & -1 \end{array}\right], \quad B=\left[\begin{array}{rr} 1 & 0 \\ 2 & -1 \\ 3 & 1 \end{array}\right]$$

Use the determinant theorems to find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}-1 & 2 & 4 \\\4 & -8 & -16 \\\3 & 0 & 5\end{array}\right]$$

Draw a sketch of the two graphs described with the indicated number of points of intersection. (There may be more than one way to do this.) A line and a circle; two points.

Graph the solution set of each system of inequalities by hand. $$\begin{array}{r} -2 < x < 2 \\ y > 1 \\ x-y > 0 \end{array}$$

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{array}{l} 2.1 x+0.5 y+1.7 z=4.9 \\ -2 x+1.5 y-1.7 z=3.1 \\ 5.8 x-4.6 y+0.8 z=9.3 \end{array}$$

Ciolino's makes dining room furniture. A buffet requires 30 hours for construction and 10 hours for finishing, a chair 10 hours for construction and 10 hours for finishing, and a table 10 hours for construction and 30 hours for finishing. The construction department has 350 hours of labor and the finishing department has 150 hours of labor available each week. How many pieces of each type of furniture should be produced each week if the factory is to run at full capacity?

Find the fourth-degree polynomial \(P(x)\) satisfying the following conditions: \(P(-2)=13\) \(P(-1)=2, P(0)=-1, P(1)=4,\) and \(P(2)=41\).

If possible, find \(A B\) and \(B A.\) $$A=\left[\begin{array}{rr} 3 & -1 \\ 1 & 0 \\ -2 & -4 \end{array}\right], \quad B=\left[\begin{array}{rrr} -2 & 5 & -3 \\ 9 & -7 & 0 \end{array}\right]$$

Use the determinant theorems to find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}6 & 8 & -12 \\\\-1 & 0 & 2 \\\4 & 0 & -8\end{array}\right]$$

Draw a sketch of the two graphs described with the indicated number of points of intersection. (There may be more than one way to do this.) A line and a parabola; no points.

Graph the solution set of each system of inequalities by hand. $$\begin{array}{c} x+y \leq 4 \\ x-y \leq 5 \\ 4 x+y \leq-4 \end{array}$$

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} 0.1 x+0.3 y+1.7 z &=0.6 \\ 0.6 x+0.1 y-3.1 z &=6.2 \\ 2.4 y+0.9 z &=3.5 \end{aligned}$$

Scheduling Production Ciolino's makes dining room furniture. A buffet requires 30 hours for construction and 10 hours for finishing, a chair 10 hours for construction and 10 hours for finishing, and a table 10 hours for construction and 30 hours for finishing. The construction department has 350 hours of labor and the finishing department has 150 hours of labor available each week. How many pieces of each type of furniture should be produced each week if the factory is to run at full capacity?

If possible, find \(A B\) and \(B A.\) $$A=\left[\begin{array}{rrr} -1 & 0 & -2 \\ 4 & -2 & 1 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & -2 \\ 5 & -1 \\ 0 & 1 \end{array}\right]$$

Use the determinant theorems to find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}4 & 8 & 0 \\\\-1 & -2 & 1 \\\2 & 4 & 3\end{array}\right]$$

Find the fifth-degree polynomial \(P(x)\) satisfying the following conditions: \(P(-2)=-8\) \(P(-1)=-1, P(0)=-4, P(1)=-5, P(2)=8,\) and \(P(3)=167\).

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} x & \leq 4 \\ x & \geq 0 \\ y & \geq 0 \\ x+2 y & \geq 2 \end{aligned}$$

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{array}{l} 53 x+95 y+12 z=108 \\ 81 x-57 y-24 z=-92 \\ -9 x+11 y-78 z=21 \end{array}$$

If possible, find \(A B\) and \(B A.\) $$A=\left[\begin{array}{rrr} 1 & -1 & 0 \\ 2 & -1 & 5 \\ 6 & 1 & -4 \end{array}\right], \quad B=\left[\begin{array}{rrr} -1 & 3 & -1 \\ 7 & -7 & 1 \end{array}\right]$$

Use the determinant theorems to find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}-4 & 1 & 4 \\\2 & 0 & 1 \\\0 & 2 & 4\end{array}\right]$$

Draw a sketch of the two graphs described with the indicated number of points of intersection. (There may be more than one way to do this.) A line and a parabola; one point.

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} 2 y+x & \geq-5 \\ y & \leq 3+x \\ x & \leq 0 \\ y & \leq 0 \end{aligned}$$

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} 103 x-886 y+431 z &=1200 \\ -55 x+981 y &=1108 \\ -327 x+421 y+337 z &=99 \end{aligned}$$

Solve each problem. A group of students bought 3 soft drinks and 2 boxes of popcorn at a movie for \(\$ 18.50 .\) A second group bought 4 soft drinks and 3 boxes of popcorn for \(\$ 26\). (a) Find a matrix equation \(A X=B\) whose solution gives the individual prices of a soft drink and a box of popcorn. Solve this matrix equation by using \(A^{-1}\) (b) Could these prices be determined if both groups had bought 3 soft drinks and 2 boxes of popcorn for \(\$ 18.50 ?\) Try to calculate \(A^{-1}\) and explain your results.

If possible, find \(A B\) and \(B A.\) $$A=\left[\begin{array}{rrr} 2 & -1 & -5 \\ 4 & -1 & 6 \\ -2 & 0 & 9 \end{array}\right], \quad B=\left[\begin{array}{rr} 1 & 2 \\ -1 & -1 \\ 2 & 0 \end{array}\right]$$

Use the determinant theorems to find each determinant. $$\operatorname{det}\left[\begin{array}{lll}6 & 3 & 2 \\\1 & 0 & 2 \\\5 & 7 & 3\end{array}\right]$$

Draw a sketch of the two graphs described with the indicated number of points of intersection. (There may be more than one way to do this.) A line and a parabola; two points.

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} 2 x+3 y & \leq 12 \\ 2 x+3 y & > -6 \\ 3 x+y & < 4 \\ x & \geq 0 \\ y & \geq 0 \end{aligned}$$

Draw a sketch of the two graphs described with the indicated number of points of intersection. (There may be more than one way to do this.) A circle and a parabola; four points.

Compare the use of an augmented matrix as a short-hand way of writing a system of linear equations with the use of synthetic division as a shorthand way to divide polynomials.

Find each matrix product if possible. $$\left[\begin{array}{rrr} 3 & -4 & 1 \\ 5 & 0 & 2 \end{array}\right]\left[\begin{array}{r} -1 \\ 4 \\ 2 \end{array}\right]$$

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

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} &y \geq 3^{x}\\\ &y \geq 2 \end{aligned}$$

Compare the use of the third type of row transformation on a matrix with the elimination method of solving a system of linear equations.

Find the equation of the parabola with vertical axis that passes through the data points shown or specified. Check your answer. $$(2,9),(-2,1),(-3,4)$$

Find each matrix product if possible. $$\left[\begin{array}{rrr} -6 & 3 & 5 \\ 2 & 9 & 1 \end{array}\right]\left[\begin{array}{r} -2 \\ 0 \\ 3 \end{array}\right]$$

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

Draw a sketch of the two graphs described with the indicated number of points of intersection. (There may be more than one way to do this.) A circle and a parabola; one point.

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} &y \leq\left(\frac{1}{2}\right)^{x}\\\ &y \geq 4 \end{aligned}$$

Solve each system graphically. Check your solutions. Do not use a calculator. $$\begin{array}{r}x+y=3 \\\2 x-y=0\end{array}$$

Find the equation of the parabola with vertical axis that passes through the data points shown or specified. Check your answer. $$(1.5,6.25),(0,-2),(-1.5,3.25)$$

Find each matrix product if possible. $$\left[\begin{array}{rr} 5 & 2 \\ -1 & 4 \end{array}\right]\left[\begin{array}{rr} 3 & -2 \\ 1 & 0 \end{array}\right]$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{array}{l}4 x+3 y=-7 \\\2 x+3 y=-11\end{array}$$

Solve each system graphically. Check your solutions. Do not use a calculator. $$\begin{array}{r}3 x-y=4 \\\x+y=0\end{array}$$

Graph the solution set of each system of inequalities by hand. $$\begin{array}{r} \ln x-y \geq 1 \\ x^{2}-2 x-y \leq 1 \end{array}$$

Find the equation of the parabola with vertical axis that passes through the data points shown or specified. Check your answer. $$(2,14),(0,0),(-1,-1)$$