To solve the given system of linear equations:
[tex]\[
\begin{cases}
x + y + z = 14 \\
2x - y + z = 17 \\
3x - z = 7 \\
\end{cases}
\][/tex]
we will find the values of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex].
### Step 1: Solve for [tex]\(z\)[/tex] from the third equation
Start with the third equation:
[tex]\[ 3x - z = 7 \][/tex]
Rearrange to solve for [tex]\(z\)[/tex]:
[tex]\[ z = 3x - 7 \][/tex]
### Step 2: Substitute [tex]\(z\)[/tex] into the first two equations
Now substitute [tex]\(z = 3x - 7\)[/tex] into the first and second equations.
#### First equation:
[tex]\[ x + y + (3x - 7) = 14 \][/tex]
Simplify:
[tex]\[ x + y + 3x - 7 = 14 \][/tex]
[tex]\[ 4x + y - 7 = 14 \][/tex]
[tex]\[ 4x + y = 21 \][/tex]
[tex]\[ y = 21 - 4x \][/tex] \quad [tex]\((1)\)[/tex]
#### Second equation:
[tex]\[ 2x - y + (3x - 7) = 17 \][/tex]
Simplify:
[tex]\[ 2x - y + 3x - 7 = 17 \][/tex]
[tex]\[ 5x - y - 7 = 17 \][/tex]
[tex]\[ 5x - y = 24 \][/tex]
[tex]\[ y = 5x - 24 \][/tex] \quad [tex]\((2)\)[/tex]
### Step 3: Equate the expressions for [tex]\(y\)[/tex]
From equations [tex]\((1)\)[/tex] and [tex]\((2)\)[/tex]:
[tex]\[ 21 - 4x = 5x - 24 \][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[ 21 + 24 = 5x + 4x \][/tex]
[tex]\[ 45 = 9x \][/tex]
[tex]\[ x = 5 \][/tex]
### Step 4: Find [tex]\(y\)[/tex] using [tex]\(x = 5\)[/tex]
Substitute [tex]\(x = 5\)[/tex] into [tex]\(y = 21 - 4x\)[/tex]:
[tex]\[ y = 21 - 4(5) \][/tex]
[tex]\[ y = 21 - 20 \][/tex]
[tex]\[ y = 1 \][/tex]
### Step 5: Find [tex]\(z\)[/tex] using [tex]\(x = 5\)[/tex]
Substitute [tex]\(x = 5\)[/tex] into [tex]\(z = 3x - 7\)[/tex]:
[tex]\[ z = 3(5) - 7 \][/tex]
[tex]\[ z = 15 - 7 \][/tex]
[tex]\[ z = 8 \][/tex]
Therefore, the solution to the system of equations is:
[tex]\[ (x, y, z) = (5, 1, 8) \][/tex]
### Step 6: Check the solution
Substitute [tex]\(x = 5\)[/tex], [tex]\(y = 1\)[/tex], and [tex]\(z = 8\)[/tex] back into the original equations:
1. [tex]\( x + y + z = 14 \)[/tex]
[tex]\[ 5 + 1 + 8 = 14 \][/tex]
[tex]\[ 14 = 14 \quad \text{(True)} \][/tex]
2. [tex]\( 2x - y + z = 17 \)[/tex]
[tex]\[ 2(5) - 1 + 8 = 17 \][/tex]
[tex]\[ 10 - 1 + 8 = 17 \][/tex]
[tex]\[ 17 = 17 \quad \text{(True)} \][/tex]
3. [tex]\( 3x - z = 7 \)[/tex]
[tex]\[ 3(5) - 8 = 7 \][/tex]
[tex]\[ 15 - 8 = 7 \][/tex]
[tex]\[ 7 = 7 \quad \text{(True)} \][/tex]
So, the solution is [tex]\( (x, y, z) = (5, 1, 8) \)[/tex].
