Question

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 \( z \).

Hint:

To solve a system of linear equations, isolate variables in terms of others and substitute into the remaining equations to find the values.

Answer 1

We are given the system of equations: \[ x + z = 5 \text{(Equation 1)}, \] \[ y + z = 7 \text{(Equation 2)}, \] \[ x + y + z = 9 \text{(Equation 3)}. \]

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: \[ x = 2, y = 4, z = 3. \]