Chapter 4

A quadrilateral has coordinates \(\left[\begin{array}{cccc}{3} & {4} & {-3} & {-4} \\ {5} & {4} & {-4} & {-3}\end{array}\right]\) a. Graph the quadrilateral. b. Find the product \(\left[\begin{array}{rr}{0} & {-1} \\ {1} & {0}\end{array}\right]\left[\begin{array}{cccc}{3} & {4} & {-3} & {-4} \\ {5} & {4} & {-4} & {-3}\end{array}\right]\) c. Graph the result as a new quadrilateral. d. What is the relationship between the quadrilaterals in parts (a) and (c)?

Write in point-slope form the equation of the line through each pair of points. $$ (0,1) \text { and }(2,-5) $$

If \(B=\left[\begin{array}{rr}{4} & {-1} \\ {2} & {0}\end{array}\right],\) what is \(\operatorname{det} B^{-1} ?\)

For Exercises \(53-56,\) use matrices \(P, Q, R, S,\) and \(I .\) Determine whether the two expressions in each pair are equal. $$P=\left[\begin{array}{ll}{3} & {4} \\ {1} & {2}\end{array}\right] \quad Q=\left[\begin{array}{rr}{-1} & {0} \\ {3} & {-2}\end{array}\right] \quad R=\left[\begin{array}{rr}{1} & {4} \\ {-2} & {1}\end{array}\right] \quad S=\left[\begin{array}{ll}{0} & {1} \\ {2} & {0}\end{array}\right] \quad I=\left[\begin{array}{ll}{1} & {0} \\ {0} & {1}\end{array}\right]$$ $$ (P+Q)(R+S) \text { and }(P+Q) R+(P+Q) S $$

Describe each translation of \(f(x)=|x|\) as vertical, horizontal, or combined. Then graph the translation. $$ f(x)=|x|-3 $$

If possible, find the dimensions of each product matrix; then find each product. If the product is not defined, explain why not. $$ \left[\begin{array}{rrr}{1} & {0} & {-5} \\ {2} & {-1} & {6}\end{array}\right]\left[\begin{array}{rrr}{2} & {4} & {-2} \\ {0} & {10} & {4} \\ {0} & {1} & {-7}\end{array}\right] $$

Write in point-slope form the equation of the line through each pair of points. $$ (-9,3) \text { and }(-4,-4) $$

For Exercises \(53-56,\) use matrices \(P, Q, R, S,\) and \(I .\) Determine whether the two expressions in each pair are equal. $$P=\left[\begin{array}{ll}{3} & {4} \\ {1} & {2}\end{array}\right] \quad Q=\left[\begin{array}{rr}{-1} & {0} \\ {3} & {-2}\end{array}\right] \quad R=\left[\begin{array}{rr}{1} & {4} \\ {-2} & {1}\end{array}\right] \quad S=\left[\begin{array}{ll}{0} & {1} \\ {2} & {0}\end{array}\right] \quad I=\left[\begin{array}{ll}{1} & {0} \\ {0} & {1}\end{array}\right]$$ $$ (P+Q)(R+S) \text { and } P R+P S+Q R+Q S $$

Describe each translation of \(f(x)=|x|\) as vertical, horizontal, or combined. Then graph the translation. $$ f(x)=|x-5|+3 $$

Write in point-slope form the equation of the line through each pair of points. $$ (1,8) \text { and }(7,2) $$

Each matrix represents the vertices of a polygon. Write a matrix to represent the vertices of the image after each transformation. \(\left[\begin{array}{rrr}{-2} & {-5} & {0} \\ {0} & {3} & {5}\end{array}\right] ;\) reflection in \(y=x\)

Which product is NOT defined? \(A .\left[\begin{array}{r}{-1} \\\ {2}\end{array}\right]\left[\begin{array}{cc}{-1} & {2}\end{array}\right]\) B. \(\left[\begin{array}{ll}{-1} & {2} \\ {-1} & {2}\end{array}\right]\left[\begin{array}{ll}{-1} & {2}\end{array}\right]\) \(C \cdot\left[\begin{array}{rr}{-1} & {2} \\ {-1} & {2}\end{array}\right]\left[\begin{array}{rr}{2} & {-1} \\ {2} & {-1}\end{array}\right]\) \(D \cdot\left[\begin{array}{ll}{-1} & {2}\end{array}\right]\left[\begin{array}{r}{-1} \\ {2}\end{array}\right]\)

Graph each equation. $$ x+y-z=5 $$

Given \(P=\left[\begin{array}{rrr}{4} & {3} & {-2} \\ {-1} & {0} & {5}\end{array}\right]\) and \(Q=\left[\begin{array}{rrr}{3} & {-2} & {-5} \\\ {-1} & {-2} & {-1}\end{array}\right],\) what is \(2 P-3 Q ?\) \(\mathrm{F} \cdot\left[\begin{array}{lll}{1} & {-5} & {3} \\ {0} & {-2} & {6}\end{array}\right]\) G. \(\left[\begin{array}{ccc}{17} & {0} & {19} \\ {-5} & {6} & {7}\end{array}\right]\) \(\mathrm{H} .\left[\begin{array}{rrr}{-1} & {12} & {11} \\ {1} & {6} & {13}\end{array}\right]\) J. \(\left[\begin{array}{lll}{1} & {5} & {3} \\ {0} & {2} & {6}\end{array}\right]\)

Graph each equation. $$ 2 x-y+3 z=12 $$

Solve \(3 Y+2\left[\begin{array}{rr}{-1} & {-3} \\ {2} & {5}\end{array}\right]=\left[\begin{array}{cc}{13} & {-9} \\ {4} & {16}\end{array}\right] .\) Show the steps of your solution.

Graph each equation. $$ -x+4 y-z=-6 $$

Solve each system by substitution. $$ \left\\{\begin{array}{l}{x=5} \\ {x-y+z=5} \\ {x+y-z=-5}\end{array}\right. $$

Given \(M=\left[\begin{array}{rr}{-3} & {4} \\ {1} & {-2}\end{array}\right]\) and \(N=\left[\begin{array}{rr}{0} & {1} \\ {-2} & {5}\end{array}\right],\) does \(M \times N=N \times M ?\) Explain.

Make a mapping diagram for each relation. Determine whether it is a function. $$ \\{(-2,4),(-1,1),(0,0),(1,1),(2,4)\\} $$

Solve each system by substitution. $$ \left\\{\begin{array}{l}{x-3 y=2 z} \\ {x+2 y-z=0} \\\ {x+y+z=10}\end{array}\right. $$

Add or subtract. $$ \left[\begin{array}{rr}{-1} & {2} \\ {0} & {17}\end{array}\right]-\left[\begin{array}{rr}{32} & {14} \\ {6} & {-10}\end{array}\right] $$

Make a mapping diagram for each relation. Determine whether it is a function. $$ \\{(27,-2),(3,-1),(0,0),(3,1),(27,2)\\} $$

Solve each system by substitution. $$ \left\\{\begin{array}{l}{3 x+3 y-z=9} \\ {2 y=x-z} \\ {x-y+5 z=9}\end{array}\right. $$

Add or subtract. $$ \left[\begin{array}{ccc}{0} & {-1} & {5} \\ {6} & {10} & {12}\end{array}\right]+\left[\begin{array}{ccc}{9} & {-5} & {7} \\ {-4} & {10} & {0}\end{array}\right] $$

Make a mapping diagram for each relation. Determine whether it is a function. $$ \\{(-2,15),(-1,1),(0,-3),(1,3),(2,15)\\} $$

a. Exercise Suppose you begin a training program by walking 2 miles every day. During the first week, the walk takes you 40 minutes per day. Each week after that, you reduce your time by one minute. Write a linear model for the number of minutes you take to walk 2 miles in week \(w .\) b. Critical Thinking Can you continue to improve at the same rate? Explain.

Graph each system of constraints. Find all vertices. Then find the values of \(x\) and \(y\) that maximize or minimize the objective function. $$ \begin{array}{c}{\left\\{\begin{array}{r}{x+y \leq 3} \\ {x \geq 0}\end{array}\right.} \\ {\text { Maximize for }} \\ {P=3 x+4 y}\end{array} $$

Graph each system of constraints. Find all vertices. Then find the values of \(x\) and \(y\) that maximize or minimize the objective function. $$ \left\\{\begin{aligned} x+2 y & \leq 8 \\ x & \geq 2 \\ y & \geq 0 \end{aligned}\right. $$ $$ \begin{array}{l}{\text { Minimize for }} \\ {C=x+3 y}\end{array} $$

Graph each system of constraints. Find all vertices. Then find the values of \(x\) and \(y\) that maximize or minimize the objective function. $$ \left\\{\begin{aligned} x+y & \leq 6 \\ 2 x+y & \leq 10 \\ x & \geq 0, y \geq 0 \end{aligned}\right. $$ $$ \begin{array}{l}{\text { Maximize for }} \\ {P=4 x+y}\end{array} $$

Graph each inequality. $$ y<4 x-1 $$

Graph each inequality. $$ y \leq-3 x+8 $$

Graph each inequality. $$ y \geq|2 x+5|-3 $$