Chapter 7

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

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

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{array}{l} 3 x+2 y+z=-7 \\ 2 x+y-z=-3 \\ -x+y+2 z=0 \end{array}$$

Three students buy different combinations of tickets for a baseball game. The first student buys 2 senior, 1 adult, and 2 student tickets for \(\$ 51 .\) The second student buys 1 adult and 5 student tickets for \(\$ 55 .\) The third student buys 2 senior, 2 adult, and 7 student tickets for \(\$ 75\). If possible, find the price of each type of ticket. Interpret your answer.

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

Draw a sketch of the two graphs described with the indicated number of points of intersection. A line and a circle; no points

Graph the solution set of each system of inequalities by hand. $$\begin{aligned}&y \leq-x^{2}\\\&y \geq x^{2}-6\end{aligned}

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

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

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

A total of \(\$ 5000\) is invested at \(2 \%, 3 \%,\) and \(4 \%\). The amount invested at \(4 \%\) equals the total amount invested at \(2 \%\) and \(3 \%\). The total interest for one year is \$145. If possible, find the amount invested at each interest rate. Interpret your answer.

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

Draw a sketch of the two graphs described with the indicated number of points of intersection. A line and a circle; one point

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

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

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

As of 2017 , the total combined number of monthly users of Facebook, Instagram, and Twitter was 223 million. The combined number of users of Instagram and Twitter was 23 million less than the number of users of Facebook. There were 18 million more users of Instagram than users of Twitter. How many users of each social network were there?

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

Draw a sketch of the two graphs described with the indicated number of points of intersection. A line and a circle; two points

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

A group of students bought 3 soft drinks and 2 boxes of popcorn at a movie for 18.50 dollar. A second group bought 4 soft drinks and 3 boxes of popcom for 26 dollar. (IMAGE CAN NOT COPY) (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 dollar Try to calculate \(A^{-1}\) and explain your results.

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

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} 2 x+y-3 z &=1 \\ x+y+2 z &=5 \\ 3 x+2 y-z &=-3 \end{aligned}$$

As of \(2017,\) the top three beauty brands on Facebook (L'Oréal, Dove, and NIVEA) had a combined 74 million followers. L'Oréal and Dove had the same number of followers, and NIVEA had 7 million fewer followers. How many followers did each brand have?

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

Draw a sketch of the two graphs described with the indicated number of points of intersection. A line and a parabola; no points

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

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

Solve each system. Round to the nearest thousandth. $$\begin{aligned} 0.07 x+0.23 y &=9 \\ -1.25 x+0.33 y &=2.4 \end{aligned}$$

The sum of the measures of the angles of any triangle is \(180^{\circ} .\) In a certain triangle, the largest angle measures \(55^{\circ}\) less than twice the medium angle, and the smallest measures \(25^{\circ}\) less than the medium angle. Find the measures of the three angles.

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

Graph the solution set of each system of inequalities by hand. $$\begin{aligned}x-y &<1 \\\\-1

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

Solve each system. Round to the nearest thousandth. $$\begin{aligned} 3 x-13 y &=17 \\ -23 x+15 y &=2 \end{aligned}$$

The perimeter of a triangle is 59 inches. The longest side is 11 inches longer than the medium side, and the medium side is 3 inches more than the shortest side. Find the length of each side.

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

Draw a sketch of the two graphs described with the indicated number of points of intersection. A line and a parabola; two points

Graph the solution set of each system of inequalities by hand. $$\begin{array}{r}x+y \leq 36 \\\\-4 \leq x \leq 4\end{array}$$

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

Solve each system. Round to the nearest thousandth. $$\begin{array}{l} 2.1 x+0.5 y+1.7 z=4.9 \\ -2 x+1.5 y-1.7 z=3.1 \\ 5.8 x-4.6 y+0.8 z=9.3 \end{array}$$

A student invested \(\$ 10,000\) in three parts. With one part, she bought mutual funds that offered a return of \(3 \%\) per year. The second part, which amounted to twice the first, was used to buy government bonds paying \(2 \%\) per year. She put the rest into a savings account that paid \(1.5 \%\) annual interest. During the first year, the total interest was \(\$ 225 .\) How much did she invest at each rate?

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

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

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

Solve each system. Round to the nearest thousandth. $$\begin{aligned} 0.1 x+0.3 y+1.7 z &=0.6 \\ 0.6 x+0.1 y-3.1 z &=6.2 \\ 2.4 y+0.9 z &=3.5 \end{aligned}$$

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?

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

Graph the solution set of each system of inequalities by hand. $$\begin{aligned}&y \geq(x-2)^{2}+3\\\&y \leq-(x-1)^{2}+6\end{aligned}$$

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

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