Exam 7: Linear Systems and Matrices

Evaluate the expression. $\left[ \begin{array} { c c } 5 & - 1 \\4 & - 4\end{array} \right] \left[ \begin{array} { c c } 0 & 3 \\- 3 & 0\end{array} \right] \left[ \begin{array} { c c } - 5 & 0 \\1 & 4\end{array} \right]$

Solve using any method. $\left\{ \begin{aligned}7 x - 9 y & = 3 \\y & = x - 4\end{aligned} \right.$

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

Perform the indicated row operations on the matrix. Show the final result. $\left[ \begin{array} { c c c } 1 & 7 & 1 \\- 7 & - 48 & - 8 \\- 5 & - 1 & - 39\end{array} \right]$ Add 7 times $R _ { 1 }$ to $R _ { 2 }$ . Add 5 times $R _ { 1 }$ to $R _ { 3 }$ .

Determine whether the two systems of linear equations yield the same solutions. If so, find the solutions using matrices. x+9y+9z =-22 y-6z =-23 z =3 x+8y-5z =-63 y+4z =7 z =3

Solve using any method. $\left\{ \begin{aligned}- 7 x - 9 y & = - 6 \\y & = x - 3\end{aligned} \right.$

Solve the system of equations graphically or algebraically. $\left\{ \begin{array} { l } y = 13 x + 24 \\y = \sqrt { 3 x + 576 }\end{array} \right.$

Write the partial fraction decomposition of the rational expression. $\frac { 5 } { x ^ { 2 } + 10 x + 16 }$

Given: $A = \left[ \begin{array} { r r r } - 7 & 3 & - 2 \\ - 6 & 6 & 5 \end{array} \right] , B = \left[ \begin{array} { r r r } 3 & - 6 & 5 \\ - 3 & 1 & 7 \end{array} \right]$ and $c = - 4$ determine $c A B ^ { 2 }$ .

Solve the system by the method of elimination. $\left\{ \begin{aligned}\frac { x + 7 } { 5 } + \frac { y - 2 } { 5 } & = - 2 \\x - y & = - 14\end{aligned} \right.$

Solve for $X$ in the equation given. $3 X = 2 A - B , A = \left[ \begin{array} { c c } - 6 & 2 \\8 & - 5\end{array} \right] \text { and } B = \left[ \begin{array} { c c } - 3 & 10 \\7 & - 1\end{array} \right]$

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $\left\{ \begin{array} { l } 7 x - 3 y - 9 z = 22 \\2 x + y + 3 z = - 3 \\2 x - 7 y - 3 z = - 17\end{array} \right.$

Given: $A = \left[ \begin{array} { r r r } - 7 & - 4 & 2 \\ 4 & - 5 & - 8 \end{array} \right] , B = \left[ \begin{array} { r r r } - 4 & 1 & - 7 \\ 8 & - 8 & 3 \end{array} \right]$ and $c = - 6$ , determine $C A B ^ { 2 }$ .

Find the equation of the circle $x ^ { 2 } + y ^ { 2 } + D x + E y + F = 0$ that passes through the points $( 1,9 ) , ( - 4,4 ) , ( 6,4 )$ .

Given: $A = \left[ \begin{array} { l l l } 4 & 2 & 5 \\1 & 2 & 1\end{array} \right] , B = \left[ \begin{array} { r r r } 5 & 5 & - 1 \\4 & 4 & 1\end{array} \right] , C = \left[ \begin{array} { r r } - 2 & - 1 \\3 & 1 \\- 2 & - 3\end{array} \right] ,$ determine $C A - C B$ .

Solve the system by the method of substitution. $\left\{ \begin{array} { l } x ^ { 2 } + y ^ { 2 } = 25 \\3 x - 4 y = 0\end{array} \right.$

Use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and solve the system. $\left\{ \begin{array} { c } x - 7 y - 3 z = 64 \\3 x + 7 y + 2 z = - 92 \\- 2 x + 14 y + 6 z = 64 \\- 8 x + 2 y + 2 z = 42\end{array} \right.$

Write the system of linear equations represented by the augmented matrix. (Use variables $x , y , z$ , and $w$ .) $\left[ \begin{array} { c c c c : c } - 4 & 0 & 0 & - 9 & - 9 \\7 & - 8 & 0 & 0 & 1 \\0 & - 4 & - 7 & - 9 & - 8 \\0 & 0 & - 4 & 1 & - 8\end{array} \right]$

Use a system of equations to find the cubic function $f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + d$ that satisfies given equations. Solve the system using matrices. f(2) =-26 f(-2) =6 f(1) =0 f(-1) =-2

Solve the system by the method of substitution. $\left\{ \begin{array} { l } y = - 3 x \\y = x ^ { 3 } + 3 x ^ { 2 } + 3 x\end{array} \right.$