Exam 10: Systems of Equations and Inequalities

Perform the row operation(s) on the given augmented matrix. -(a) $R _ { 2 } = - 2 r _ { 1 } + r _ { 2 }$ (b) $\mathrm { R } _ { 3 } = - 2 \mathrm { r } _ { 1 } + \mathrm { r } _ { 3 }$ (c) $\mathrm { R } _ { 3 } = 6 \mathrm { r } _ { 2 } + \mathrm { r } _ { 3 }$ $\left[ \begin{array} { r r r | r } 1 & - 3 & - 5 & - 2 \\2 & - 5 & - 4 & 5 \\2 & 5 & 4 & 6\end{array} \right]$

Graph the system of inequalities. - $\left\{\begin{array}{l}x^{2}+y^{2} \leq 100 \\-7 x+3 y \leq-21\end{array}\right.$ A) B) C) D)

Graph the inequality. - $x+y<-3$ A) B) C) D)

Solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent. - $\left\{ \begin{aligned}3 x + 9 y - z & = 53 \\x - 9 y + 9 z & = 11 \\- 4 x + y + z & = 5\end{aligned} \right.$

Write a system of linear inequalities that has the given graph. - A) $\left\{\begin{array}{r}y \geq 0 \\x \geq 0 \\x \leq 4 \\y+x \leq 8\end{array}\right.$ B) $\left\{\begin{array}{r}y \geq 0 \\x \geq 0 \\x \leq 4 \\y+x \geq 8\end{array}\right.$ C) $\left\{\begin{array}{r}y \geq 0 \\x \geq 0 \\x \leq 8 \\y+x \leq 4\end{array}\right.$ D) $\left\{\begin{array}{l}x \leq 4 \\y+x \leq 8\end{array}\right.$

Use a graphing utility to solve the system of equations. Express the solution rounded to two decimal places. - $\left\{ \begin{array} { r } x ^ { 3 } + y ^ { 2 } = 2 \\x ^ { 2 } y = 4\end{array} \right.$

Write the augmented matrix for the system. - $\left\{ \begin{array} { r } 8 x - 2 z = 60 \\9 y + 7 z = 33 \\2 x - 2 y + 9 z = 74\end{array} \right.$

Graph the system of inequalities. - 2x+3y\leq6 x-y\leq3 x\geq1 A) B) C) D)

Solve the problem. -A person with no more than $3,000 to invest plans to place the money in two investments, telecommunications and pharmaceuticals. The telecommunications investment is to be no more than 4 times the pharmaceuticals Investment. Write a system of inequalities to describe the situation. Let x = amount to be invested in Telecommunications and y = amount to be invested in pharmaceuticals. A) x+y=3,000 y\geq4x x\geq0 y\geq0 B) x+y\leq3,000 4x\leqy x\geq0 y\geq0 C) x+y\leq3,000 x\leq4y x\geq0 y\geq0 D) x+y=3,000 x\leq4y x\geq0 y\geq0

Solve for x. - $\left| \begin{array} { l l } 8 & x \\2 & 5\end{array} \right| = 32$

Graph the region indicated by graphing the system of inequalities. Label all points of intersection. - $\left\{ \begin{array} { l } y \geq x ^ { 2 } - 8 \\y \leq - x ^ { 2 }\end{array} \right.$ A) $\mathrm { O } ( 0,0 ) , \mathrm { A } ( 0 , - 8 )$ B) $\mathrm { O } ( 0,0 ) , \mathrm { A } ( 0 , - 8 )$ C) $\mathrm { O } ( 0,0 ) , \mathrm { A } ( 0 , - 8 )$ D) $\mathrm { O } ( 0,0 ) , \mathrm { A } ( 0 , - 8 )$

Solve the system of equations. - $\left\{\begin{array}{r}x+4 y-z=3 \\x+5 y-2 z=5 \\3 x+12 y-3 z=9\end{array}\right.$

Use the properties of determinants to find the value of the second determinant, given the value of the first. - $\text { Given } \left| \begin{array} { l l l } \mathrm { s } & \mathrm { t } & \mathrm { u } \\\mathrm { v } & \mathrm { w } & \mathrm { x } \\4 & 2 & 8\end{array} \right| = 3 \text {, find the value of } \left| \begin{array} { c c c } 32 - \mathrm { s } & 16 - \mathrm { t } & 64 - \mathrm { u } \\\mathrm { v } & \mathrm { w } & \mathrm { x } \\4 & 2 & 8\end{array} \right|$

Solve the problem. -A right triangle has an area of 33 square inches. The square of the hypotenuse is 157. Find the lengths of the legs of the triangle. Round your answer to the nearest inch.

Show that the matrix has no inverse. - $\mathrm { A } = \left[ \begin{array} { r r } 14 & - 6 \\- 7 & 3\end{array} \right]$

Perform the row operation(s) on the given augmented matrix. - =4+ -7 -5 -1 -10 6 -2 9 5 28 -6 6 18

Write the partial fraction decomposition of the rational expression. - $\frac { 8 x - 47 } { ( x + 2 ) ( x - 5 ) }$

Write the system of equations associated with the augmented matrix. Do not solve. - $\left[ \begin{array} { r r r | r } 8 & 8 & - 8 & 7 \\6 & 6 & 0 & 9 \\3 & 0 & - 4 & - 6\end{array} \right]$

Solve the system. - $\left\{ \begin{array} { l } 7 x - 2 y = 5 \\28 x - 8 y = 15\end{array} \right.$

Use Cramer's rule to solve the linear system. - $\left\{ \begin{array} { l } 3 x + 3 y = 51 \\2 x - 2 y = 2\end{array} \right.$