Question

solve the system.\

$$\begin{cases}-3x + 6y - 2z = 5\\\\7x + 2y + 9z = 19\\\\-x + y - z = 0\\end{cases}$$

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

Explanation:

Step1: Simplify the third equation

From \(-x + y - z = 0\), we can express \(y\) as \(y=x + z\).

Step2: Substitute \(y = x + z\) into the first two equations

• Substitute into the first equation \(-3x + 6y - 2z = 5\):

\(-3x+6(x + z)-2z = 5\)
Expand: \(-3x + 6x+6z - 2z = 5\)
Simplify: \(3x + 4z = 5\) (Equation 4)

• Substitute into the second equation \(7x + 2y + 9z = 19\):

\(7x+2(x + z)+9z = 19\)
Expand: \(7x + 2x+2z + 9z = 19\)
Simplify: \(9x + 11z = 19\) (Equation 5)

Step3: Solve the system of Equation 4 and Equation 5

From Equation 4: \(3x=5 - 4z\), so \(x=\frac{5 - 4z}{3}\)
Substitute \(x=\frac{5 - 4z}{3}\) into Equation 5:
\(9\times\frac{5 - 4z}{3}+11z = 19\)
Simplify: \(3(5 - 4z)+11z = 19\)
Expand: \(15-12z + 11z = 19\)
Simplify: \(15 - z = 19\)
Solve for \(z\): \(-z=19 - 15=4\), so \(z=- 4\)

Step4: Find \(x\) using \(z = - 4\)

Substitute \(z=-4\) into Equation 4: \(3x+4\times(-4)=5\)
\(3x-16 = 5\)
\(3x=5 + 16=21\)
\(x = 7\)

Step5: Find \(y\) using \(x = 7\) and \(z=-4\)

Since \(y=x + z\), \(y=7+(-4)=3\)

Answer:

\((7, 3, -4)\)