Exam 7: Linear Systems and Matrices

Determine which one of the ordered triples below is a solution of the given system of equations. $\left\{ \begin{array} { l } 8 x + 6 y - 4 z = 74 \\x - 5 y + 8 z = - 102 \\5 x - 3 y + 2 z = - 46\end{array} \right.$

An object moving vertically is at the given heights at the specified times. Find the position equation $s = \frac { 1 } { 2 } a t ^ { 2 } + v _ { o } t + s _ { o }$ for the object. At $t = 1$ second, $s = 198$ feet At $t = 2$ seconds, $s = 182$ feet At $t = 3$ seconds, $s = 134$ feet

Write the augmented matrix for the system of linear equations. $\left\{ \begin{aligned}x + 4 y - z & = - 9 \\- 9 y + 4 z & = - 6 \\x + 5 z & = - 4\end{aligned} \right.$

Solve using any method. $\left\{ \begin{aligned}7 x + 6 y & = 15 \\y & = x - 7\end{aligned} \right.$

Use a determinant to determine whether the points below are collinear. $( 3 , - 1 ) , ( 0 , - 3 ) , ( 24,13 )$

Solve the system of linear equations. $\left\{ \begin{aligned}2 x - 2 y + 3 z & = - 21 \\2 x + z & = - 11 \\- 2 x + 4 y - 4 z & = 28\end{aligned} \right.$

An augmented matrix that represents a system of linear equations (in variables $x , y$ , and $z$ ) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. $\left[ \begin{array} { c c c : c } 1 & 0 & 0 & 9 \\0 & 1 & 0 & 8 \\0 & 0 & 1 & - 7\end{array} \right]$

Write the partial fraction decomposition of the improper rational expression. $\frac { x ^ { 3 } - 4 x ^ { 2 } + 3 x + 2 } { x ^ { 2 } + x - 2 }$

Solve the system by the method of elimination. $\left\{ \begin{array} { c } 5 x - y = 11 \\- 7 x - y = - 25\end{array} \right.$

Find the minor $M _ { 13 }$ and its cofactor $C _ { 13 }$ of the matrix $\left[ \begin{array} { c c c } - 9 & 6 & - 24 \\ 9 & - 6 & 18 \\ - 3 & 9 & - 18 \end{array} \right]$ .

Solve the system of linear equations. $\left\{ \begin{aligned}x + y + z + w & = 11 \\- 2 x - y - 3 z + 4 w & = - 5 \\- 2 x - 5 z - w & = - 23 \\3 x + 5 y + z - 2 w & = 16\end{aligned} \right.$

Solve the system by the method of elimination. Round numbers to three decimal places. $\left\{ \begin{array} { c } 5.7 x + 7.1 y = 3.2 \\- 0.8 x - 4.8 y = - 1.1\end{array} \right.$

Evaluate the determinant $\left| \begin{array} { c c } 7 x & x \ln x \\ 7 & 2 + \ln x \end{array} \right|$ in which the entries are functions.

Find $| A B |$ , if $A = \left[ \begin{array} { r r r } 5 & - 2 & - 2 \\- 3 & 4 & - 1 \\- 5 & 6 & 2\end{array} \right] , B = \left[ \begin{array} { r r r } 1 & - 1 & - 3 \\- 5 & - 3 & 4 \\- 5 & - 1 & - 2\end{array} \right]$

Solve the system of linear equations $\left\{ \begin{array} { l } 4 x _ { 1 } - 8 x _ { 2 } - 4 x _ { 3 } - 8 x _ { 4 } = 0 \\12 x _ { 1 } - 20 x _ { 2 } - 8 x _ { 3 } - 12 x _ { 4 } = - 9 \\8 x _ { 1 } - 20 x _ { 2 } - 8 x _ { 3 } - 20 x _ { 4 } = 6 \\- 4 x _ { 1 } + 16 x _ { 2 } + 16 x _ { 3 } + 44 x _ { 4 } = 0\end{array} \right.$ using the inverse matrix $\frac { 1 } { 4 } \left[ \begin{array} { c c c c } - 24 & 7 & 1 & - 2 \\ - 10 & 3 & 0 & - 1 \\ - 29 & 7 & 3 & - 2 \\ 12 & - 3 & - 1 & 1 \end{array} \right]$ .

Solve the system of equations algebraically. $\left\{ \begin{array} { r } x y + 25 = 0 \\49 x - 100 y - 700 = 0\end{array} \right.$

Fill in the blank using elementary row operations to form a row-equivalent matrix. 2 -1 -9 8 -6 4 2 -1 -9 0 \square 40

Use a system of equations to find the specified equation that passes through the points. Solve the system using matrices. Parabola: $y = a x ^ { 2 } + b x + c$

Find the minor $M _ { 13 }$ and its cofactor $C _ { 13 }$ of the matrix $\left[ \begin{array} { c c c } 9 & - 6 & 24 \\ - 9 & 6 & - 18 \\ 3 & - 9 & 18 \end{array} \right]$ .

Evaluate the expression. $\left[ \begin{array} { c } - 5 \\- 1 \\5\end{array} \right] \left( \left[ \begin{array} { l l } 9 & - 7\end{array} \right] + \left[ \begin{array} { l l } - 7 & - 8\end{array} \right] \right)$