If \[ \begin{bmatrix} x + z \\ y + z \\ x + y + z \end{bmatrix} = \begin{bmatrix} 5 \\ 7 \\ 9 \end{bmatrix}, \] then find the value of \( x \), \( y \), and \( z \).
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Solution and Explanation
Step 1: Solve for \( x \) and \( y \) in terms of \( z \).
From Equation 1:
\[
x = 5 - z.
\]
From Equation 2:
\[
y = 7 - z.
\]
Step 2: Substitute \( x \) and \( y \) into Equation 3.
Substitute \( x = 5 - z \) and \( y = 7 - z \) into Equation 3:
\[
(5 - z) + (7 - z) + z = 9.
\]
Simplify the equation:
\[
5 + 7 - z - z + z = 9 $\Rightarrow$ 12 - z = 9 $\Rightarrow$ z = 3.
\]
Step 3: Find \( x \) and \( y \).
Now that we know \( z = 3 \), substitute this value into the expressions for \( x \) and \( y \):
\[
x = 5 - 3 = 2,
\]
\[
y = 7 - 3 = 4.
\]
Conclusion:
The values of \( x \), \( y \), and \( z \) are:
\[
x = 2, y = 4, z = 3.
\]
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