Exam 6: Systems and Matrices

Solve the problem using matrices. -A chemist has prepared two acid solutions, one of which is $6 \%$ acid by volume, another $10 \%$ acid. How many cubic centimeters of each should the chemist mix together to obtain $50 \mathrm {~cm} ^ { 3 }$ of a solution that is $8.16 \%$ acid.

Find the partial fraction decomposition for the rational expression. $\frac { 4 x ^ { 3 } + 8 x ^ { 2 } + 5 x - 7 } { 2 x ^ { 2 } - x - 1 }$

Solve the equation for x. - $\left| \begin{array} { l l } x & 3 \\2 & 1\end{array} \right| = 12$

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set. x+y=1 x+y=5

Write the augmented matrix for the system. Do not - 6x+2z=10 4y+7z=26 9x+3y+7z=32

Perform the operation or operations when possible. - $\left[ \begin{array} { r r } - 1 & 1 \\2 & 5\end{array} \right] + \left[ \begin{array} { r r } 6 & 2 \\5 & - 2\end{array} \right]$

Decide whether or not the matrices are inverses of each other. - $\left[ \begin{array} { r r r } 2 & - 1 & 0 \\- 1 & 1 & - 2 \\1 & 0 & - 1\end{array} \right] \text { and } \left[ \begin{array} { r r r } 1 & - 1 & 2 \\- 3 & - 2 & 4 \\- 1 & 1 & 1\end{array} \right]$

Give all solutions of the nonlinear system of equations, including those with nonreal complex components. - $x ^ { 2 } + y ^ { 2 } = 113$ $x - y = 1$

Find the inverse, if it exists, for the matrix. - $\left[ \begin{array} { r r } 2 & - 3 \\ - 4 & 6 \end{array} \right]$

Provide an appropriate response. -Fill in the blank to complete the statement. Two matrices are equal if they have the same _______and if all corresponding elements are equal.

Write the augmented matrix for the system. Do not solve the system. 7x+3z-2=0 2y+6z-20=0 3x+9y+2z-12=0

Write the word or phrase that best completes each statement or answers the question. Which method should be used to solve the system? Explain your answer, including a description of the first step. 2+ =1 -7+5 =16

Provide an appropriate response. -For a certain system of three linear equations, the solution set is $\{ 5 , - 19 , - 2 \}$ , and $D = 13$ . Find the values of $D _ { X }$ , $\mathrm { D } _ { \mathrm { y } }$ , and $\mathrm { D } _ { \mathrm { Z } }$ .

Find the value of the determinant. $\left| \begin{array} { l l } 0 & - 1 \\8 & 0\end{array} \right|$

Provide an appropriate response. -You are graphing $2 x ^ { 2 } + 3 y ^ { 2 } > 5$ . You have drawn the graph of $2 x ^ { 2 } + 3 y ^ { 2 } = 5$ . What single point will allow you to determine the correct region?

Use a graphing calculator to Express solutions with approximations to the nearest thousandth. - 1.2x+y=-8 -4.5x-y=10

Solve the system by elimination. - x+5y=-5 -7x+6y=35

Solve the system. x-y+3z=1 4x+z=0 x+5y+z=-5

Find the inverse, if it exists, for the matrix. - $\left[ \begin{array} { l l l } 1 & 3 & 2 \\1 & 3 & 3 \\2 & 7 & 8\end{array} \right]$

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set. x+y+z=-3 x-y+2z=2 4x+y+z=9