Exam 15: Systems of Equations: Matrices and Determinants

Solve the system. $\left( \begin{array} { c } x - 3 y + 5 z = 44 \\5 x + 2 y + 3 z = 25 \\3 x - 2 y - 3 z = 7\end{array} \right)$

Evaluate $4 \times 4$ determinant. $\left| \begin{array} { c c c c } 5 & - 1 & 3 & 5 \\1 & 0 & 3 & 1 \\3 & 5 & 0 & 1 \\8 & 3 & 6 & - 8\end{array} \right|$ Use the properties of determinants to your advantage.

The matrix is in reduced echelon form. $\left. \left[ \begin{array} { l l l } 1 & 1 & 0 \\0 & 1& 2\\0 &0 & 1\end{array} \right] \quad \begin{array} { r} -4 \\7 \\-1\end{array} \right]$

The matrix is in reduced echelon form. $\left. \left[ \begin{array} { l l } 1 & 0 \\0 & 1\end{array} \right] \quad \begin{array} { l l } 13 \\3\end{array} \right]$

Solve the system. $\left( \begin{array} { r l } x &+& 2 y &-& z & =& 9 \\x &+& 3 y &+& 2 z & =& 10 \\- x &-& 3 y &-& 2 z & =& - 10\end{array} \right)$

Indicate whether the matrix is in reduced echelon form. $\left. \left[ \begin{array} { l l } 1 & 0 \\0 & 1\end{array} \right] \quad \begin{array} { l } 9 \\5\end{array} \right]$ Enter yes or no .

A matrix of dimension $3 \times 4$ has 3 rows and 4 columns.

Solve the system by the substitution or elimination method. $\left( \begin{array} { r l } 2 x^{2}-3 y^{2}=-1 \\2 x^{2}+3 y^{2}=5\end{array} \right)$

Solve the system. $\left( \begin{array} { r l r } - 2 x + 3 y + 5 z & = 23 \\3 x - 5 z & = - 13 \\5 z & = 25\end{array} \right)$

Solve the problem by using a system of equations. The sum of the digits of a two-digit number is 17. If the digits are reversed, the newly formed number is 9 larger than the original number. Find the original number.

Give a step-by-step explanation of how to evaluate the determinant. $\left| \begin{array} { c c c } 5 & 0 & 4 \\1 & - 4 & 6 \\7 & 0 & 9\end{array} \right|$

A small company makes three different types of bird houses. Each type requires the services of three different departments, as indicated by the following table. Type A Type B Type C Cutting department 0.2 hour 0.1 hour 0.3 hour Finishing department 0.5 hour 0.5 hour 0.4 hour Assembly department 0.1 hour 0.1 hour 0.3 hour The cutting, finishing, and assembly departments have available a maximum of 61, 128, and 52 work-hours per week, respectively. How many bird houses of each type should be made per week so that the company is operating at full capacity?

The elimination-by-addition method can be used to solve systems of nonlinear equations.

The solution set is infinitely many ordered triples if the three planes have a common line of intersection..

Graph the system so that approximate real number solutions (if there are any) can be predicted. $\left( \begin{array} { l } y = x ^ { 2 } - 2 \\x + y = - 3\end{array} \right)$

Use a matrix approach to solve the system. $\left( \begin{array} { r } x_{1}&+&3 x_{2}&-&x_{3}&+&2 x_{4} & =&-6 \\2 x_{1}&+&4 x_{2}&+&2 x_{3}&-&x_{4} & =&-5 \\-3 x_{1}&-&5 x_{2}&+&3 x_{3}&+&x_{4} & =&7 \\4 x_{1}&+&7 x_{2}&-&2 x_{3}&-&3 x_{4} & =&-8\end{array} \right)$ If a system is inconsistent, so indicate. In those cases enter inconsistent . Otherwise, enter your answer in the form ( x , y ).

Use a matrix approach to solve the system. $\left( \begin{array} { c c c c } x - 5 y & = & 15 \\5 x + 4 y & = & - 41\end{array} \right)$ If a system is inconsistent, so indicate. In those cases enter inconsistent . Otherwise, enter your answer in the form ( x , y ).

The matrix is the reduced echelon matrix for a system with variables x 1 , x 2 , x 3 , and x 4 . Find the solution set of the system. $\left[ \begin{array} { r r r r | r } 1 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & - 4 \\0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 1 & - 1\end{array} \right]$ If a system is inconsistent, so indicate.

Solve the system. $\left( \begin{array} { l } 5 x - 6 y + 3 z = 17 \\5 x + 6 y - 4 z = 26 \\4 x - 2 y + 2 z = 12\end{array} \right)$

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