Exam 10: Systems of Equations and Inequalities

Write the word or phrase that best completes each statement or answers the question. -Find real numbers $a , b$ , and c such that the graph of the function $y = a x ^ { 2 } + b x + c$ contains the points $( - 2 , - 4 )$ , $( 1$ , $- 1 )$ , and $( 3 , - 19 )$ .

Choose the one alternative that best completes the statement or answers the question. - $\frac { 3 x ^ { 3 } + 4 x ^ { 2 } } { \left( x ^ { 2 } + 5 \right) ^ { 2 } }$

Perform the indicated matrix operations. -Let $A = \left[ \begin{array} { r r r } - 6 & - 6 & - 3 \\ - 3 & 2 & - 8 \\ 5 & - 2 & - 7 \end{array} \right]$ and $B = \left[ \begin{array} { r r r } - 5 & - 5 & - 1 \\ 7 & - 6 & 1 \\ 1 & - 5 & - 2 \end{array} \right]$ . Find 2A - 2B.

Find the inverse of the matrix. - $\left[ \begin{array} { l l l } 1 & 0 & 0 \\5 & 1 & 0 \\0 & 7 & 1\end{array} \right]$

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

2x + 3y ≥6 - 2x+3y\leq6 x-y\leq3 x\geq1

Write the partial fraction decomposition of the rational expression. - $\frac { x + 1 } { ( x - 2 ) ^ { 2 } ( x + 4 ) }$

Verify that the values of the variables listed are solutions of the system of equations. - $\left\{ \begin{array} { r } x - y + 4 z = - 3 \\3 x + z = - 2 \\x + 2 y + z = - 12 \\\end{array} \right.$ x = 0 , y = - 5 , z = - 2

Write the word or phrase that best completes each statement or answers the question. Set up the linear programming problem. -The Jillson's have up to $75,000 to invest. They decide that they want to have at least $40,000 invested in stable bonds yielding 6% and that no more than $20,000 should be invested in more volatile bonds yielding 12%. (a) Using x to denote the amount of money invested in the stable bonds and y the amount invested in the more volatile bonds, write a system of linear inequalities that describe the possible amounts of each investment. (b) Graph the system and label the corner points.

Use the properties of determinants to find the value of the second determinant, given the value of the first. - $\left| \begin{array} { r r r } \mathrm { x } & \mathrm { y } & \mathrm { z } \\\mathrm { u } & \mathrm { v } & \mathrm { w } \\1 & - 2 & - 4\end{array} \right| = 119 \left| \begin{array} { c c c } 1 & - 2 & - 4 \\\mathrm { u } & \mathrm { v } & \mathrm { w } \\\mathrm { x } & \mathrm { y } & \mathrm { z }\end{array} \right| = ?$

Solve the system. - $\left\{ \begin{aligned}8 x + y & = 2 \\- 24 x - 3 y & = - 6\end{aligned} \right.$

Solve the system of equations. - $\left\{ \begin{array} { r } x - y + 3 z = 10 \\2 x + z = 4 \\x + 2 y + z = 8\end{array} \right.$

2x + 3y ≥6 - + \leq36 +\geq4

2x + 3y ≥6 - $\left\{ \begin{aligned}x - y & \leq 3 \\y & \leq 2\end{aligned} \right.$

Choose the one alternative that best completes the statement or answers the question. -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 $\mathrm { y } =$ amount to be invested in pharmaceuticals.

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

Choose the one alternative that best completes the statement or answers the question. Use Cramer's rule to solve the linear system. - $\left\{ \begin{array} { r } - 2 x + 4 y = - 12 \\ - 2 x + 5 y = - 11 \\ \end{array} \right.$

Solve the problem. -The Family Arts Center charges $22 for adults, $12 for senior citizens, and $7 for children under 12 for their live performances on Sunday afternoon. This past Sunday, the paid revenue was $10,134 for 757 tickets sold. There Were 41 more children than adults. How many children attended?

Find the inverse of the matrix. - $\left[ \begin{array} { l l l } 1 & 0 & 8 \\1 & 2 & 3 \\2 & 5 & 3\end{array} \right]$

Solve the system of equations using elimination. - 2-4=2 3+2=35