Question

solve the following system.\

$$\begin{cases}2x + y = 2\\\\-3x + z = -7\\\\-4y + 3z = 22\\end{cases}$$

\
\
x = \quad y = \quad z =

Explanation:

Step1: Solve for y from first equation

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

Step2: Solve for z from second equation

From \(-3x + z = -7\), we get \(z = 3x - 7\).

Step3: Substitute y and z into third equation

Substitute \(y = 2 - 2x\) and \(z = 3x - 7\) into \(-4y + 3z = 22\):
\[

$$\begin{align*} -4(2 - 2x) + 3(3x - 7) &= 22\\ -8 + 8x + 9x - 21 &= 22\\ 17x - 29 &= 22\\ 17x &= 22 + 29\\ 17x &= 51\\ x &= \frac{51}{17}\\ x &= 3 \end{align*}$$

\]

Step4: Find y using x = 3

Substitute \(x = 3\) into \(y = 2 - 2x\):
\(y = 2 - 2\times3 = 2 - 6 = -4\)

Step5: Find z using x = 3

Substitute \(x = 3\) into \(z = 3x - 7\):
\(z = 3\times3 - 7 = 9 - 7 = 2\)

Answer:

\(x = 3\)
\(y = -4\)
\(z = 2\)