Exam 10: Systems of Equations and Inequalities

Solve the system of equations $\left\{ \begin{array} { r } x + 3 y - 2 z = 1 \\2 x + 4 y + z = - 4 \\2 x + 6 y - z = 2\end{array} \right.$ by converting to a matrix equation and using its inverse coefficient matrix $\left[ \begin{array} { c c c } \frac { 5 } { 3 } & \frac { 3 } { 2 } & - \frac { 11 } { 6 } \\\\- \frac { 2 } { 3 } & - \frac { 1 } { 2 } & \frac { 5 } { 6 } \\\\- \frac { 2 } { 3 } & 0 & \frac { 1 } { 3 }\end{array} \right]$ .

Graph the inequality $7 x + 3 y \leq - 1$ .

Solve the system of linear equations. $\left\{ \begin{array} { r } 5 x + 3 y + 11 z = 37 \\3 y + z = 12 \\5 x + 9 y + 13 z = 61\end{array} \right.$

Given $A = \left[ \begin{array} { c c c } - 2 & 0 & 3 \\0 & - 4 & 4 \\1 & 3 & - 2\end{array} \right]$ and $B = \left[ \begin{array} { c c c } 0 & 2 & - 2 \\5 & - 3 & 2 \\1 & - 2 & 0\end{array} \right]$ , find $A B$ , or explain why the operation cannot be performed.

Find the complete solution of the linear system or show that it is inconsistent. $\left\{ \begin{array} { c } x + y + z = 0 \\y - 3 z = 1 \\2 x + y + 4 z = 0\end{array} \right.$

Find the partial fraction decomposition of $\frac { 3 } { ( x - 2 ) ( x + 3 ) }$ .

Find all solutions for the system. $\left\{ \begin{array} { l } - \frac { 2 } { x } - \frac { 3 } { y } = 1 \\\frac { 2 } { x } + \frac { 5 } { y } = 1\end{array} \right.$

State whether the system of equations is linear. $\left\{ \begin{array} { r } x + 2 y + z = 3 \\\sqrt { 2 } x + y - \sqrt { 3 } z = 1 \\x + y + 4 z = - 4\end{array} \right.$

Solve for $x$ . $\left| \begin{array} { c c c } x & - 4 & 10 \\0 & x - 1 & 5 \\0 & 0 & x + 2\end{array} \right| = 0$

State whether the equation $3 \sqrt { 5 } x - \frac { 3 y } { 5 } + z = 0$ is linear.

Use back-substitution to solve the triangular system. $\left\{ \begin{aligned}x + y - z & = - 5 \\y - z & = 5 \\\frac { 1 } { 3 } z & = - \frac { 4 } { 3 }\end{aligned} \right.$

Find the quadratic polynomial $y = a x ^ { 2 } + b x + c$ whose graph passes through the points $( - 3,7 )$ , $( - 2,1 )$ , and $( 1,7 )$ .

Given $A = \left[ \begin{array} { c c } 2 & - 3 \\4 & 2\end{array} \right]$ and $B = \left[ \begin{array} { c c } - 1 & 5 \\0 & 25\end{array} \right]$ , find $A - B$ , or explain why the operation cannot be performed.

Solve the system. $\left\{ \begin{aligned}x + y & = 8 \\- x + y & = 0\end{aligned} \right.$

Find the complete solution of the linear system or show that it is inconsistent. $\left\{ \begin{array} { r } x + 2 y + z + w = 0 \\x + 2 y - z - w = 0 \\x + 4 y + z + 2 w = - 1 \\x + 3 y + z + w = 2\end{array} \right.$

Solve for $x$ and $y$ . $\left[ \begin{array} { l l } 1 & 2 \\3 & 4\end{array} \right] \left[ \begin{array} { l } x \\y\end{array} \right] = \left[ \begin{array} { l } 1 \\2\end{array} \right]$

Find all solutions for the system. $\left\{ \begin{array} { l } x = y ^ { 2 } \\x - y = 2\end{array} \right.$

A man invests a total of $\$ 21,000$ in two accounts, one paying $5.5 \%$ simple interest per year and the other paying $8.5 \%$ simple interest per year. His annual interest from both accounts is $\$ 1$ , $280$ . How much did he invest at each rate?

Use the graphical method to find all the solutions of the system of equations. $\left\{ \begin{array} { l } \frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 25 } = 1 \\y = 4 x ^ { 2 } - 4\end{array} \right.$

Use a graphing device to find all solutions of the system of equations. $\left\{ \begin{array} { l } x ^ { 2 } + y ^ { 2 } = 4 \\y = - 2 x + 1\end{array} \right.$