Chapter 7

Solve each system. Round to the nearest thousandth. $$\begin{array}{l} 53 x+95 y+12 z=108 \\ 81 x-57 y-24 z=-92 \\ -9 x+11 y-78 z=21 \end{array}$$

Doctor Rug sells rug-cleaning machines. The EZ model weighs 10 pounds and comes in a 10-cubic-foot box. The compact model weighs 20 pounds and comes in an 8 -cubic-foot box. The commercial model weighs 60 pounds and comes in a 28 -cubic-foot box. Each of the company's delivery vans has 248 cubic feet of space and can hold a maximum of 440 pounds. Find all possible ways in which the three types of boxes can be loaded into the van so that the boxes require the maximum space of 248 cubic feet and weigh exactly 440 pounds. Assume that a fraction of a box cannot be loaded.

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

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

Graph the solution set of each system of inequalities by hand. $$\begin{array}{r}x \geq 0 \\\x+y \leq 4 \\\2 x+y \leq 5\end{array}$$

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

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

Solve each system. Round to the nearest thousandth. $$\begin{aligned} 103 x-886 y+431 z &=1200 \\ -55 x+981 y &=1108 \\ -327 x+421 y+337 z &=99 \end{aligned}$$

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

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

Graph the solution set of each system of inequalities by hand. $$\begin{aligned}3 x-2 y & \geq 6 \\\x+y & \leq-5 \\\y & \leq 4\end{aligned}$$

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

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

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

The table shows the selling prices for three representative homes. Price \(P\) is given in thousands of dollars, age \(A\) in years, and home size \(S\) in thousands of square feet. These data may be modeled by the equation \(P=a+b A+c S\). $$\begin{array}{c|c|c} \text { Price (P) } & \text { Age (A) } & \text { Size (S) } \\ \hline 190 & 20 & 2 \\ 320 & 5 & 3 \\ 50 & 40 & 1 \end{array}$$ (a) Write a system of linear equations whose solution gives \(a, b,\) and \(c\) (b) Solve this system of linear equations. (c) Predict the price of a home that is 10 years old and has 2500 square feet.

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

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

Graph the solution set of each system of inequalities by hand. $$\begin{array}{l}-2

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

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

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

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. $$\begin{array}{r}-21 \\\x-y>0\end{array}$$

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 \(A^{-1}\).

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

Solve each system. Write solutions in terms of \(z\) if necessary. $$\begin{array}{r} x-3 y+2 z=10 \\ 2 x-y-z=15 \end{array}$$

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

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

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

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 \(A\)

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

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. $$\left[\begin{array}{lll}0 & 3 & -4\end{array}\right]\left[\begin{array}{rrr}-2 & 6 & 3 \\ 0 & 4 & 2 \\ -1 & 1 & 4\end{array}\right]$$

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

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

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 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. $$(1.5,6.25),(0,-2),(-1.5,3.25)$$

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

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

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

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. $$(2,14),(0,0),(-1,-1)$$

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

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

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

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{array}{l}5 x-y=-4 \\\3 x+2 z=4 \\\4 y+3 z=22\end{array}$$

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