Question

solve the system.\

$$\begin{cases}2x + 3y - 2z = -7\\\\x - 2y + 4z = 15\\\\2y + z = 1\\end{cases}$$

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

Explanation:

Step1: Solve the third equation for z

From \(2y + z = 1\), we get \(z = 1 - 2y\).

Step2: Substitute z into the second equation

Substitute \(z = 1 - 2y\) into \(x - 2y + 4z = 15\):
\(x - 2y + 4(1 - 2y)=15\)
\(x - 2y + 4 - 8y = 15\)
\(x - 10y + 4 = 15\)
\(x - 10y = 11\) (Equation A)

Step3: Substitute z into the first equation

Substitute \(z = 1 - 2y\) into \(2x + 3y - 2z = -7\):
\(2x + 3y - 2(1 - 2y)= -7\)
\(2x + 3y - 2 + 4y = -7\)
\(2x + 7y - 2 = -7\)
\(2x + 7y = -5\) (Equation B)

Step4: Solve Equation A for x

From Equation A: \(x = 11 + 10y\)

Step5: Substitute x into Equation B

Substitute \(x = 11 + 10y\) into Equation B:
\(2(11 + 10y)+7y = -5\)
\(22 + 20y + 7y = -5\)
\(22 + 27y = -5\)
\(27y = -27\)
\(y = -1\)

Step6: Find x using y = -1

Substitute \(y = -1\) into \(x = 11 + 10y\):
\(x = 11 + 10(-1)=11 - 10 = 1\)

Step7: Find z using y = -1

Substitute \(y = -1\) into \(z = 1 - 2y\):
\(z = 1 - 2(-1)=1 + 2 = 3\)

Answer:

\((1, -1, 3)\)