Exam 15: Systems of Equations: Matrices and Determinants

Evaluate $3 \times 3$ determinant. $\left| \begin{array} { c c c } 1 & 4 & - 1 \\5 & 1 & 4 \\4 & 6 & 5\end{array} \right|$ Use the properties of determinants to your advantage. Please enter your answer as a number.

Use a matrix approach to solve the system. $\left( \begin{array} { r l l l } 2 x - 5 y & = & 21 \\x + 2 y & = & - 3\end{array} \right)$ If a system is inconsistent, so indicate. In those cases enter inconsistent . Otherwise, enter your answer in the form ( x , y ).

Solve the system. $\left( \begin{array} { c } 4 x + 7 y - 7 z = 22 \\2 x - 2 y + 6 z = - 12 \\5 x - 4 y + 7 z = - 7\end{array} \right)$

Consider the following matrix. $A = \left[ \begin{array} { c c c c } 1 & 6 & 8 & 10 \\- 5 & 7 & 1 & 5 \\7 & 10 & 12 & 4 \\6 & 5 & - 1 & 9\end{array} \right]$ Form matrix B by interchanging rows 1 and 3 of matrix A . Now use your calculator to show that $| B | = - | A |$ .

Solve the system. $\left( \begin{array} { c } 2 x& - &y &+& z& =& - 6 \\3 x& -& 2 y &+& 5 z& =& - 17 \\3 x &+ &y &-& 5 z& = &9\end{array} \right)$

Solve the system by the substitution or elimination method. $\left( \begin{array} { l } y = x ^ { 2 } + 6 x + 4 \\2 x + y = - 3\end{array} \right)$ Express your answer as an ordered pair. If there is more than one solution, separate your answers with a comma.

Use a matrix approach to solve the system. $\left(\begin{array}{c}x&-&4 y&-&z&=&9 \\6 x&+&y&-&3 z&=&-34 \\-5 x&+&7 y&+&2 z&=&-5\end{array}\right)$ If a system is inconsistent, so indicate. In those cases enter inconsistent . Otherwise, enter your answer in the form ( x , y ).

If A is a $4 \times 4$ matrix with integer entries, the cofactor of each element is an integer.

Solve the system. $\left(\begin{array}{rll}x+2 y-5 z & = & -20 \\3 y-z & = & -1 \\3 y+3 z & = & 15\end{array}\right)$

Solve the problem by using a system of equations. The tens digit of a two-digit number is 7 less than three times the units digit. If the sum of the digits is 9, find the number.

Solve each system of equations by the substitution or elimination method. Match each system of equations with the corresponding answer. - $( 5 , - 20 )$

Solve the homogeneous system. $\left( \begin{array} { r } 3 x + y + z = 0 \\x - y + 2 z = 0 \\4 x - 5 y - 2 z = 0\end{array} \right)$

Solve the system by the substitution or elimination method. $\left( \begin{array} { r l } x ^ { 2 } + y ^ { 2 } & = 17 \\x + 4 y & = 17\end{array} \right)$

Solve the system. $\left( \begin{array} { lllll} - 4 x &+ &5 y &+& 2 z & =& 0 \\&&4 x &-& 3 z & = & - 24 \\&&&&4 z & = & 16\end{array} \right)$

Solve the system by the substitution or elimination method. $\left( \begin{array} { l } y = - x ^ { 2 } + 2 \\y = x ^ { 2 } - 1\end{array} \right)$

Evaluate $2 \times 2$ determinant. $\left| \begin{array} { c c } - 3 & 1 \\9 & 7\end{array} \right|$

A gift store is making a mixture of almonds, pecans, and peanuts, which sell for $2.50 per pound, $4.00 per pound, and $1.00 per pound, respectively. The storekeeper wants to make 20 pounds of the mix to sell at $1.90 per pound. The number of pounds of peanuts is to be three times the number of pounds of pecans. Find the number of pounds of each to be used in the mixture. Please enter your answer as an ordered triple ( x , y , z ), where x , y , z are the numbers of pounds of almonds, pecans, and peanuts, respectively.

If $D = 3$ and $D _ { x } = 0$ , then the system of equations is inconsistent.

One day last summer, Jim went kayaking on the Little Susitna River in Alaska. Paddling upstream against the current, he traveled 12 miles in 3 hours. Then he turned around and paddled twice as fast downstream and, with the help of the current, traveled 17 miles in 1 hour. Find the rate of the current. __________ miles per hour

A video store rents new release movies for $5 and favorites for $2.65. One day the number of new release movies rented was twice the number of favorites. If the total income for that day was $822.25, how many movies of each kind were rented? __________ new release __________ favorites