Question

solve the system.\

$$\begin{cases}-5x - 6y = -17\\\\-3x - 5y + 5z = 2\\\\-6x - 5y + z = -13\\end{cases}$$

\
enter your answer as an ordered triple (x, y, z).\
(\boxed{?}, \boxed{?}, \boxed{?})

Explanation:

Step1: Solve the first equation for x

From \(-5x - 6y = -17\), we can rewrite it as:
\(-5x = 6y - 17\)
\(x=\frac{17 - 6y}{5}\)

Step2: Eliminate z from the second and third equations

The second equation: \(-3x - 5y + 5z = 2\)
The third equation: \(-6x - 5y + z = -13\), multiply it by 5: \(-30x - 25y + 5z = -65\)
Subtract the second equation from this new equation:
\((-30x - 25y + 5z)-(-3x - 5y + 5z)=-65 - 2\)
\(-27x - 20y=-67\)

Step3: Substitute x into the new equation

Substitute \(x = \frac{17 - 6y}{5}\) into \(-27x - 20y=-67\):
\(-27\times\frac{17 - 6y}{5}-20y=-67\)
Multiply both sides by 5:
\(-27(17 - 6y)-100y=-335\)
\(-459 + 162y-100y=-335\)
\(62y=459 - 335\)
\(62y = 124\)
\(y = 2\)

Step4: Find x using y

Substitute \(y = 2\) into \(x=\frac{17 - 6y}{5}\):
\(x=\frac{17-6\times2}{5}=\frac{17 - 12}{5}=1\)

Step5: Find z using x and y

Substitute \(x = 1\) and \(y = 2\) into the third equation \(-6x - 5y + z=-13\):
\(-6\times1-5\times2+z=-13\)
\(-6 - 10+z=-13\)
\(z=-13 + 16\)
\(z = 3\)

Answer:

\((1, 2, 3)\)