Exam 15: Systems of Equations: Matrices and Determinants

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

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. $\left( \begin{array} { c c c c } t & =& 9 u & + & 13 \\7 t & - &7 u & = & - 77\end{array} \right)$ 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 as an ordered pair ( t , u ).

Solve the system by using the substitution method. $\left( \begin{array} { l l l l } x& +& y & = & 4 \\y & = &x & + & 6\end{array} \right)$ 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 as an ordered pair ( x , y ).

Solve the problem by using a system of equations. The tens digit of a two-digit number is 17 less than three times the units digit. If the sum of the digits is 11, find the number.

Solve the system by the substitution or elimination method. $\left( \begin{array} { r l } 2 x ^ { 2 } + y ^ { 2 } & = 15 \\x ^ { 2 } - y ^ { 2 } & = 6\end{array} \right)$

Use a matrix approach to solve the system. $\left( \begin{array} { c c c c } x _ { 1 } &+& 6 x _ { 2 } &-& x _ { 3 } &+& 5 x _ { 4 } &= &- 4 \\5 x _ { 1 } &+& 3 x _ { 2 } &+& 5 x _ { 3 } &-& x _ { 4 } &= &- 9 \\- 6 x _ { 1 } &-& 4 x _ { 2 } &+& 6 x _ { 3 } &+& x _ { 4 } &= &23 \\ 7 x _ { 1 } &+& 9 x _ { 2 } &-& 5 x _ { 3 } &-& 6 x _ { 4 } &= &- 34 \\\end{array} \right)$

A box contains $6.80 in nickels, dimes, and quarters. There are 40 coins in all, and the sum of the numbers of nickels and dimes is two less than the number of quarters. How many coins of each kind are there?

Solve the system. $\left(\begin{array}{rl}2 x+4 y-z & =-20 \\4 x-5 y+5 z & =53 \\3 x+y-2 z & =-7\end{array}\right)$

Solve the system by the substitution or elimination method. $\left( \begin{array} { c } x + y = - 5 \\x ^ { 2 } + 2 y ^ { 2 } - 4 y - 97 = 0\end{array} \right)$

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. $\left[ \begin{array} { l l l l | r } 1 & 8 & 0 & 0 & 5 \\0 & 0 & 1 & 0 & - 13 \\0 & 0 & 0 & 1 & 5 \\0 & 0 & 0 & 0 & 0\end{array} \right]$

Use the appropriate property of determinants from this section to justify the true statement. $\left| \begin{array} { c c c } 10 & 2 & 3 \\5 & - 1 & 6 \\20 & - 4 & 12\end{array} \right| = 15 \left| \begin{array} { c c c } 2 & 2 & 1 \\1 & - 1 & 2 \\4 & - 4 & 4\end{array} \right| = 60 \left| \begin{array} { c c c } 2 & 2 & 1 \\1 & - 1 & 2 \\1 & - 1 & 1\end{array} \right|$ 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.

A system of two linear equations can be solved by graphing the lines on the same set of axes.

Evaluate $4 \times 4$ determinant. $\left| \begin{array} { c c c c } 1 & 3 & 6 & 8 \\- 7 & 4 & 0 & 9 \\- 4 & 6 & 3 & 8 \\3 & 1 & 5 & 4\end{array} \right|$ Use the properties of determinants to your advantage. Please enter your answer as a number.

In a system of three linear equations, if two of the planes coincide the solution is infinitely many solutions.

Solve the system. $\left( \begin{array} { r l l } 5 x &+ &3 y& - &2 z & = & - 4 \\4 x& -& 7 y &+&7 z & = & 68 \\2 x &-& 5 y &+ &3 z & = & 36\end{array} \right)$

Solve the system by the substitution or elimination method. $\left( \begin{array} { r l l } 5 x ^ { 2 } - 6 y ^ { 2 } & = & - 1 \\5 x ^ { 2 } + 6 y ^ { 2 } & = & 11\end{array} \right)$ 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. $\left( \begin{array} { c l c c c } 5 x&-&4 y&-&5 z & =&3 \\x&+&4 y&+&5 z & =&9 \\-x&+&8 y&-&10 z & =&-4\end{array} \right)$ If the equations of the system are dependent, or if a system is inconsistent, so indicate. In those cases enter dependent or inconsistent .

Use the appropriate property of determinants from this section to justify the true statement. $\left| \begin{array} { c c c } 12 & 3 & 3 \\4 & - 1 & 9 \\20 & - 5 & 15\end{array} \right| = 12 \left| \begin{array} { c c c } 3 & 3 & 1 \\1 & - 1 & 3 \\5 & - 5 & 5\end{array} \right| = 60 \left| \begin{array} { c c c } 3 & 3 & 1 \\1 & - 1 & 3 \\1 & - 1 & 1\end{array} \right|$ Do not evaluate the determinants.

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. $\left( \begin{array} { llll } 4 x &-& 7 y & = & 6 \\8 x &-& 14 y & =& - 5 \end{array} \right)$ 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 as an ordered pair ( x , y ).

Use a matrix approach to solve the system. $\left( \begin{array} { l } x_{1}&-&4 x_{2}&-&3 x_{3}&+&x_{4} & =&10 \\-3 x_{1}&+&3 x_{2}&+&x_{3}&-&3 x_{4} & =&-5 \\4 x_{1}&-&3 x_{2}&-&4 x_{3}&+&4 x_{4} & =&11 \\7 x_{1}&+&x_{2}&+&6 x_{3}&-&3 x_{4} & =&-33\end{array} \right)$ If a system is inconsistent, so indicate. In those cases enter inconsistent . Otherwise, enter your answer in the form ( x , y ).