Exam 15: Systems of Equations: Matrices and Determinants

Solve the system by the substitution or elimination method. $\left( \begin{array} { r l } 4 x ^ { 2 } + 9 y ^ { 2 } & = 25 \\2 x + 3 y & = 7\end{array} \right)$ Express your answer as an ordered pair. If there is more than one solution, separate your answers with a comma.

Use Cramer s rule to find the solution set for the system. $\left( \begin{array} { r l l } 3 x - 3 y & = & 12 \\x & = & - 1\end{array} \right)$ If the equations of the system are dependent, or if a system is inconsistent, so indicate. In those cases enter dependent or inconsistent . Otherwise, enter your answer in the form ( x , y ).

Use a matrix approach to solve the system. $\left( \begin{array} { c c c c } x - 4 y & = & 11 \\4 x + 3 y & = & - 32\end{array} \right)$

Solve the system. $\left(\begin{array}{llll}x&-&5 y&+&2 z & = & 24 \\2 x&+&2 y&+&4 z & = & 12 \\2 x&-&2 y&-&3 z & = & -4\end{array}\right)$

Solve the system by the substitution or elimination method. $\left( \begin{array} { r l } 9 x ^ { 2 } + 4 y ^ { 2 } & = 25 \\3 x + 2 y & = 7\end{array} \right)$

Solve the problem by using a system of equations. The sum of the digits of a two-digit number is 7. If the digits are reversed, the newly formed number is 45 larger than the original number. Find the original number.

Indicate whether the matrix is in reduced echelon form. $\left. \left[ \begin{array} { l l l l } 1 & 0 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 0 & 1\end{array} \right] \quad \begin{array} { r } - 2 \\10\\-9\\-3\end{array} \right]$ Enter yes or no .

Use the appropriate property of determinants from this section to justify the true statement. $\left| \begin{array} { c c c } 7 & 9 & 15 \\8 & - 12 & 4 \\7 & 6 & - 1\end{array} \right| = - \left| \begin{array} { c c c } 7 & 15 & 9 \\8 & 4 & - 12 \\7 & - 1 & 6\end{array} \right|$ Do not evaluate the determinants. Property number 1: If any row (or column) of a square matrix A contains only zeros, then | A | = 0. Property number 2: If square matrix B is obtained from square matrix A by interchanging two rows (or two columns), then | B | = $-$ | A |. Property number 3: If square matrix B is obtained from square matrix A by multiplying each element of any row (or column) of A by some real number k , then | B | = k | A |. Property number 4: If square matrix B is obtained from square matrix A by adding k times a row (or column) of A to another row (or column) of A , then | B | = | A |. Property number 5: If two rows (or columns) of a square matrix A are identical, then | A | = 0. Enter property number only.

Cramer's rule is a method of solving a system of equations by using determinants.

Solve the system. $\left( \begin{array} { r l } 5 x& + &5 y &- &3 z & =& - 14 \\&&2 y &+& 4 z & = &20 \\&&3 y &- &2 z & =& 6\end{array} \right)$

Solve the system by the substitution or elimination method. $\left( \begin{array} { l } y = x ^ { 2 } + 3 x - 2 \\y = x ^ { 2 } + 5 x + 6\end{array} \right)$ 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. $\left( \begin{array} { r l } x - 3 y - 6 z & = - 20 \\6 x + 7 y - z & = - 45 \\5 x + y + 5 z & = 11\end{array} \right)$ If a system is inconsistent, so indicate. In those cases enter inconsistent .

Give a step-by-step description of how you would solve the system. $\left( \begin{array} { c } 2 x - y + 3 z = 10 \\x - 2 y - z = 4 \\3 x + 5 y + 8 z = 12\end{array} \right)$

To solve the system $\left( \begin{array} { r } 2 x - y + 3 z = 5 \\2 y + z = 3 \\4 z = 8\end{array} \right)$ , solve the equation $4 z = 8$ first.

Use Cramer s rule to find the solution set for the system. $\left( \begin{array} { r l l } 2 x + 2 y & = 2 \\x & = 3\end{array} \right)$

Use the appropriate property of determinants from this section to justify the true statement. $\left| \begin{array} { c c c } 6 & 9 & 11 \\8 & - 10 & 3 \\6 & 5 & - 1\end{array} \right| = - \left| \begin{array} { c c c } 6 & 11 & 9 \\8 & 3 & - 10 \\6 & - 1 & 5\end{array} \right|$ Do not evaluate the determinants.

Solve the system by the substitution or elimination method. $\left( \begin{array} { l } y& = &- x ^ { 2 } &+ &6 \\x &+& y &= &7\end{array} \right)$

Solve the system. $\left( \begin{array} { rrrrrr } 4 x &+ &3 y& -& z & = &- 8 \\&&3 x &-& 5 y & = &12 \\&&4 x &+& 5 y & = & - 19\end{array} \right)$

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. $\left[ \begin{array} { r r r r | r } 1 & 6 & 0 & 0 & 10 \\0 & 0 & 1 & 0 & - 5 \\0 & 0 & 0 & 1 & 9 \\0 & 0 & 0 & 0 & 0\end{array} \right]$ If a system is inconsistent, so indicate.

Solve the homogeneous system. $\left( \begin{array} { r } 4 x + y + z = 0 \\x - y + 3 z = 0 \\5 x - 5 y - 3 z = 0\end{array} \right)$ If the equations of the system are dependent, or if a system is inconsistent, so indicate. In those cases enter dependent or inconsistent .