Chapter 4

Business Your friend's mother plans to open a restaurant. The initial investment is \(\$ 90,000 .\) Weekly expenses will be about \(\$ 8200 .\) If the weekly income is about \(\$ 8900,\) in how many weeks will she get back her investment?

Solve using Cramer's Rule. (Hint: Start by substituting \(m=\frac{1}{x}\) and \(n=\frac{1}{y}\) .) $$ \left\\{\begin{array}{l}{\frac{4}{x}-\frac{2}{y}=1} \\\ {\frac{10}{x}+\frac{20}{y}=0}\end{array}\right. $$

For Exercises \(38-45,\) use matrices \(D, E,\) and \(F\) shown below. Perform the indicated operations if they are defined. If an operation is not defined, label it undefined. $$D=\left[\begin{array}{rrr}{1} & {2} & {-1} \\ {0} & {3} & {1} \\ {2} & {-1} & {-2}\end{array}\right] \quad E=\left[\begin{array}{rrr}{2} & {-5} & {0} \\\ {1} & {0} & {-2} \\ {3} & {1} & {1}\end{array}\right] \quad F=\left[\begin{array}{rr}{-3} & {2} \\ {-5} & {1} \\ {2} & {4}\end{array}\right]$$ $$ D(E F) $$

Find the constant of variation for a direct variation that includes the given values. \((3,5)\)

Find the slope and \(y\) -intercept of the graph of each function. $$ y=2 x-6 $$

Solve each system. $$ \left\\{\begin{aligned} w+x+y+z &=3 \\\\-w+x-2 y+z &=-2 \\ 2 x-y+z &=1 \\ w &+y-z=& 2 \end{aligned}\right. $$

Solve each matrix equation. $$ \left[\begin{array}{rr}{1} & {9} \\ {6} & {-6}\end{array}\right]=\left[\begin{array}{rr}{-7} & {-9} \\ {4} & {5}\end{array}\right] X+\left[\begin{array}{rr}{3} & {4} \\ {4} & {-3}\end{array}\right] $$

For Exercises \(38-45,\) use matrices \(D, E,\) and \(F\) shown below. Perform the indicated operations if they are defined. If an operation is not defined, label it undefined. $$D=\left[\begin{array}{rrr}{1} & {2} & {-1} \\ {0} & {3} & {1} \\ {2} & {-1} & {-2}\end{array}\right] \quad E=\left[\begin{array}{rrr}{2} & {-5} & {0} \\\ {1} & {0} & {-2} \\ {3} & {1} & {1}\end{array}\right] \quad F=\left[\begin{array}{rr}{-3} & {2} \\ {-5} & {1} \\ {2} & {4}\end{array}\right]$$ $$ D-2 E $$

Find the constant of variation for a direct variation that includes the given values. \(\left(\frac{1}{2}, 9\right)\)

Find the slope and \(y\) -intercept of the graph of each function. $$ 3 y=6+2 x $$

Solve each system. $$ \left\\{\begin{aligned} 2 x+2 y+z &=4 \\ w &+y-z=-2 \\ w+x+y+z &=3 \\\\-4 w &+z=& 2 \end{aligned}\right. $$

Solve each matrix equation. $$ \left[\begin{array}{ll}{4} & {7} \\ {1} & {2}\end{array}\right] X+\left[\begin{array}{rr}{2} & {7} \\ {-3} & {4}\end{array}\right]=\left[\begin{array}{rr}{6} & {2} \\ {-2} & {3}\end{array}\right] $$

Which matrix equation represents the system \(\left\\{\begin{array}{c}{2 x-3 y=-3} \\ {-5 x+y=14}\end{array} ?\right.\) A. \(\left[\begin{array}{r}{x} \\\ {y}\end{array}\right]\left[\begin{array}{rr}{2} & {-3} \\ {-5} & {1}\end{array}\right]=\left[\begin{array}{c}{-3} \\ {14}\end{array}\right]\) B. \(\left[\begin{array}{rr}{2} & {-3} \\ {-5} & {1}\end{array}\right]\left[\begin{array}{l}{x} \\\ {y}\end{array}\right]=\left[\begin{array}{c}{-3} \\ {14}\end{array}\right]\) c. \(\left[\begin{array}{rr}{2} & {-3} \\ {-5} & {1}\end{array}\right]\left[\begin{array}{c}{-3} \\\ {14}\end{array}\right]=\left[\begin{array}{l}{x} \\ {y}\end{array}\right]\) D. \(\left[\begin{array}{c}{-3} \\\ {14}\end{array}\right]\left[\begin{array}{ll}{x} & {y}\end{array}\right]=\left[\begin{array}{rr}{2} & {-3} \\ {-5} & {1}\end{array}\right]\)

For Exercises \(38-45,\) use matrices \(D, E,\) and \(F\) shown below. Perform the indicated operations if they are defined. If an operation is not defined, label it undefined. $$D=\left[\begin{array}{rrr}{1} & {2} & {-1} \\ {0} & {3} & {1} \\ {2} & {-1} & {-2}\end{array}\right] \quad E=\left[\begin{array}{rrr}{2} & {-5} & {0} \\\ {1} & {0} & {-2} \\ {3} & {1} & {1}\end{array}\right] \quad F=\left[\begin{array}{rr}{-3} & {2} \\ {-5} & {1} \\ {2} & {4}\end{array}\right]$$ $$ (E-D) F $$

In each relation, \(y\) varies directly as \(x .\) Find \(y\) when \(x=9\). \(y=6\) when \(x=4\)

Find the slope and \(y\) -intercept of the graph of each function. $$ -x-2 y=12 $$

For Exercises \(38-45,\) use matrices \(D, E,\) and \(F\) shown below. Perform the indicated operations if they are defined. If an operation is not defined, label it undefined. $$D=\left[\begin{array}{rrr}{1} & {2} & {-1} \\ {0} & {3} & {1} \\ {2} & {-1} & {-2}\end{array}\right] \quad E=\left[\begin{array}{rrr}{2} & {-5} & {0} \\\ {1} & {0} & {-2} \\ {3} & {1} & {1}\end{array}\right] \quad F=\left[\begin{array}{rr}{-3} & {2} \\ {-5} & {1} \\ {2} & {4}\end{array}\right]$$ $$ (D D) E $$

In each relation, \(y\) varies directly as \(x .\) Find \(y\) when \(x=9\). \(y=8\) when \(x=4\)

Find the slope and \(y\) -intercept of the graph of each function. $$ y=5 x $$

The perimeter of the rectangle at the right is 28 \(\mathrm{cm} .\) The perimeter of each of the triangles is 24 \(\mathrm{cm} .\) The diagonal of the rectangle is 2 \(\mathrm{cm}\) longer than the longer side of the rectangle. a. Write a system of three equations in three unknowns. b. Simplify the system to a system of two equations in two unknowns. c. Write an augmented matrix for the system in part (b). d. Find the dimensions of the rectangle. e. Find the length of the diagonal.

For Exercises \(38-45,\) use matrices \(D, E,\) and \(F\) shown below. Perform the indicated operations if they are defined. If an operation is not defined, label it undefined. $$D=\left[\begin{array}{rrr}{1} & {2} & {-1} \\ {0} & {3} & {1} \\ {2} & {-1} & {-2}\end{array}\right] \quad E=\left[\begin{array}{rrr}{2} & {-5} & {0} \\\ {1} & {0} & {-2} \\ {3} & {1} & {1}\end{array}\right] \quad F=\left[\begin{array}{rr}{-3} & {2} \\ {-5} & {1} \\ {2} & {4}\end{array}\right]$$ $$ (2 D)(3 F) $$

Critical Thinking Suppose \(A=\left[\begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right]\) . For what values of \(a, b, c,\) and \(d\) will \(A\) be its own inverse? \((\text { Hint: There is more than one correct answer.) }\)

Each matrix represents the vertices of a polygon. Write a matrix to represent the vertices of the image after each transformation. $$ \left[\begin{array}{rrr}{-3} & {0.5} & {-5} \\ {0} & {3} & {3}\end{array}\right] ; \text { dilation of } 2 $$

Use the system $$\left\\{\begin{array}{ll}{2 x+y-3 z=} & {-2} \\ {4 x-3 y+6 z=} & {9 \text { for Exercises } 48 \text { and } 49} \\ {-2 x-2 y+9 z=} & {7}\end{array}\right.$$ What is the value of the determinant \(D_{y} ?\) \(\begin{array}{lllll}{\text { A. }-36} & {\text { B. }-24} & {\text { C. }-18} & {\text { D. } 36}\end{array}\)

How can you write the three equations at the right as a matrix equation for a system? Explain your steps. $$ \begin{array}{l}{2 x-3 y+z+10=0} \\ {x+4 y=2 z+11} \\ {-2 y+3 z+7=3 x}\end{array} $$

Each matrix represents the vertices of a polygon. Write a matrix to represent the vertices of the image after each transformation. $$\left[\begin{array}{cccc}{17} & {6} & {6} & {2} \\ {5} & {10} & {2} & {6}\end{array}\right] ; \text { reflection in } y=x$$

Use the system $$\left\\{\begin{array}{ll}{2 x+y-3 z=} & {-2} \\ {4 x-3 y+6 z=} & {9 \text { for Exercises } 48 \text { and } 49} \\ {-2 x-2 y+9 z=} & {7}\end{array}\right.$$ What is the value of the determinant \(D_{Z} ?\) \(\begin{array}{lllll}{\text { F. }-36} & {\text { G. }-24} & {\text { H. }-18} & {} & {\text { 1. } 36}\end{array}\)

Critical Thinking Explain why a \(2 \times 3\) matrix does not have a multiplicative inverse.

Evaluate the determinant of each matrix.$$ \left[\begin{array}{rrr}{-1} & {3} & {7} \\ {5} & {-4} & {-2} \\ {0} & {2} & {10}\end{array}\right] $$

Writing Suppose \(A\) is a \(2 \times 3\) matrix and \(B\) is a \(3 \times 2\) matrix. Are \(A B\) and \(B A\) equal? Explain your reasoning. Include examples.

Solve each system of equations by using the inverse of the coefficient matrix. $$\left\\{\begin{aligned} x+4 y+3 z &=3 \\ 2 x-5 y-z &=5 \\ 3 x+2 y-2 z &=-3 \end{aligned}\right. $$

Let \(M=\left[\begin{array}{cc}{a} & {b} \\ {c} & {d}\end{array}\right]\) and \(N=\left[\begin{array}{cc}{e} & {f} \\ {g} & {h}\end{array}\right]\) Prove that the product of the determinants of \(M\) and \(N\) equals the determinant of the matrix product \(M N .\)

Evaluate the determinant of each matrix. $$ \left[\begin{array}{ccc}{17} & {0} & {0} \\ {0} & {17} & {0} \\ {0} & {0} & {17}\end{array}\right] $$

Solve each system of equations by using the inverse of the coefficient matrix. $$ \left\\{\begin{aligned} x+y+z &=-1 \\ y+3 z &=-5 \\ x &+z=-2 \end{aligned}\right. $$

Evaluate the determinant of each matrix. $$ \left[\begin{array}{rrr}{-3} & {0} & {5} \\ {5} & {-3} & {2} \\ {-3} & {-5} & {-2}\end{array}\right] $$

What are the coordinates of \(X(5,1), Y(-5,-3),\) and \(Z(-1,3)\) reflected in the line \(y=x ?\) $$ \begin{array}{ll}{\text { A. } X^{\prime}(-5,-1), Y^{\prime}(5,3), Z^{\prime}(1,-3)} & {\text { B. } X^{\prime}(1,5), Y^{\prime}(-3,-5), Z^{\prime}(3,-1)} \\ {\text { C. } X^{\prime}(-1,-5), Y^{\prime}(3,5), Z^{\prime}(-3,1)} & {\text { D. } X^{\prime}(5,1), Y^{\prime}(-5,-3), Z^{\prime}(-1,3)}\end{array} $$

Solve for \(x\) and \(y\) $$ \left[\begin{array}{rr}{2 x} & {1} \\ {2} & {0}\end{array}\right]\left[\begin{array}{rr}{1} & {3} \\ {2} & {-y}\end{array}\right]=\left[\begin{array}{rr}{-4} & {-9} \\ {2} & {6}\end{array}\right] $$

Solve each system of inequalities by graphing. $$ \left\\{\begin{array}{l}{2 x+y<3} \\ {-x-y \geq 1}\end{array}\right. $$

What is the determinant of \(\left[\begin{array}{rr}{-2} & {-3} \\ {5} & {0}\end{array}\right] ?\)

Add or subtract. $$ \left[\begin{array}{cc}{5} & {-3} \\ {4} & {11}\end{array}\right]+\left[\begin{array}{rr}{4} & {0} \\ {-9} & {1}\end{array}\right] $$

Reflection in which line takes the figure with vertices \(A(0,0), B(-2,4)\) , \(c(-4,2),\) and \(D(-3,0)\) to \(A^{\prime}(0,0), B^{\prime}(-2,-4), C^{\prime}(-4,-2),\) and \(D^{\prime}(-3,0) ?\) F. \(x\) -axis \(\quad\) G. \(y\) -axis \(\quad\) H. \(y=x \quad\) J. \(y=-x\)

Solve for \(x\) and \(y\) $$ \left[\begin{array}{rr}{2 x} & {1} \\ {2} & {0}\end{array}\right]\left[\begin{array}{rr}{0} & {3} \\ {2 x} & {-y}\end{array}\right]=\left[\begin{array}{rr}{-4} & {-9} \\ {0} & {6}\end{array}\right] $$

Solve each system of inequalities by graphing. $$ \left\\{\begin{array}{lll}{2 x} & { \leq} & {0} \\ {-x+y} & {>} & {-1}\end{array}\right. $$

What is the determinant of \(\left[\begin{array}{cc}{\frac{3}{10}} & {\frac{1}{5}} \\ {\frac{1}{8}} & {\frac{1}{3}}\end{array}\right] ?\) Enter your answer as a fraction.

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

Add or subtract. $$ \left[\begin{array}{rrr}{-1} & {2} & {0} \\ {10} & {-5} & {15} \\ {17} & {3} & {-4}\end{array}\right]-\left[\begin{array}{rrr}{0} & {6} & {-3} \\ {4} & {-7} & {11} \\ {-9} & {10} & {-1}\end{array}\right] $$

Solve each system of inequalities by graphing. $$ \left\\{\begin{aligned} x &<3 \\ y & \geq-4 \\\\-x+y &<5 \end{aligned}\right. $$

If \(A=\left[\begin{array}{rr}{4} & {2} \\ {-3} & {-1}\end{array}\right],\) and the inverse of \(A\) is \(x\left[\begin{array}{rr}{-1} & {-2} \\ {3} & {4}\end{array}\right],\) what is the value of \(x\) ? Enter your answer as a fraction.

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

Describe each translation of \(f(x)=|x|\) as vertical, horizontal, or combined. Then graph the translation. $$ f(x)=|x+4| $$