Question

solve the system.
\

$$\begin{cases}6x + y + 6z = 7 7x - y + 5z = 8 x - 2y - z = 0 \\end{cases}$$

select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
\bigcirc a. there is one solution. the solution set is \\{ (\square, \square, \square) \\}.
(simplify your answers.)
\bigcirc b. there are infinitely many solutions. the system is dependent.
\bigcirc c. there is no solution.

Explanation:

Step1: Eliminate \( y \) from first two equations

Add the first equation \( 6x + y + 6z = 7 \) and the second equation \( 7x - y + 5z = 8 \):
\( (6x + 7x) + (y - y) + (6z + 5z) = 7 + 8 \)
\( 13x + 11z = 15 \) --- Equation (4)

Step2: Eliminate \( y \) from first and third equations

Multiply the first equation \( 6x + y + 6z = 7 \) by 2: \( 12x + 2y + 12z = 14 \)
Add it to the third equation \( x - 2y - z = 0 \):
\( (12x + x) + (2y - 2y) + (12z - z) = 14 + 0 \)
\( 13x + 11z = 14 \) --- Equation (5)

Step3: Analyze Equation (4) and (5)

Equation (4): \( 13x + 11z = 15 \)
Equation (5): \( 13x + 11z = 14 \)
Subtract Equation (5) from Equation (4):
\( (13x - 13x) + (11z - 11z) = 15 - 14 \)
\( 0 = 1 \) (which is a contradiction)

Answer:

C. There is no solution.