To solve the system of equations:
[tex]\[
\begin{aligned}
1) \quad x - 3y - z &= -3, \\
2) \quad -x &= 8, \\
3) \quad 8y - 4z &= 0, \\
4) \quad 2x - 15y + 7z &= -15,
\end{aligned}
\][/tex]
we need to find the values of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex] that satisfy all four equations simultaneously.
### Step-by-Step Solution
1) Solve for [tex]\(x\)[/tex] using Equation (2):
[tex]\[
-x = 8 \implies x = -8
\][/tex]
2) Substitute [tex]\(x = -8\)[/tex] into Equations (1), (3), and (4):
Substitute [tex]\(x = -8\)[/tex] into Equation (1):
[tex]\[
-8 - 3y - z = -3 \implies -3y - z = -3 + 8 \implies -3y - z = 5 \\
\implies z = -3y - 5 \quad \text{(Equation A)}
\][/tex]
As Equation (3) does not contain [tex]\(x\)[/tex], it remains unchanged:
[tex]\[
8y - 4z = 0
\][/tex]
Substitute [tex]\(x = -8\)[/tex] into Equation (4):
[tex]\[
2(-8) - 15y + 7z = -15 \implies -16 - 15y + 7z = -15 \\
\implies 7z - 15y = 1 \quad \text{(Equation B)}
\][/tex]
3) Solve for [tex]\(y\)[/tex] and [tex]\(z\)[/tex] using Equations (3) and (B):
From Equation (3):
[tex]\[
8y - 4z = 0 \implies 4z = 8y \implies z = 2y
\][/tex]
Substitute [tex]\(z = 2y\)[/tex] into Equation (B):
[tex]\[
7(2y) - 15y = 1 \implies 14y - 15y = 1 \implies -y = 1 \implies y = -1
\][/tex]
4) Find [tex]\(z\)[/tex] using [tex]\(y = -1\)[/tex]:
Substitute [tex]\(y = -1\)[/tex] back into [tex]\(z = 2y\)[/tex]:
[tex]\[
z = 2(-1) \implies z = -2
\][/tex]
We have now determined that:
[tex]\[
x = -8, \quad y = -1, \quad z = -2
\][/tex]
### Verification
Let us verify that these values satisfy all original equations:
1. For [tex]\(x - 3y - z = -3\)[/tex]:
[tex]\[
-8 - 3(-1) - (-2) = -8 + 3 + 2 = -3
\][/tex]
2. For [tex]\(-x = 8\)[/tex]:
[tex]\[
-(-8) = 8
\][/tex]
3. For [tex]\(8y - 4z = 0\)[/tex]:
[tex]\[
8(-1) - 4(-2) = -8 + 8 = 0
\][/tex]
4. For [tex]\(2x - 15y + 7z = -15\)[/tex]:
[tex]\[
2(-8) - 15(-1) + 7(-2) = -16 + 15 - 14 = -15
\][/tex]
The values [tex]\(x = -8\)[/tex], [tex]\(y = -1\)[/tex], and [tex]\(z = -2\)[/tex] satisfy all the equations. Therefore, the solution to the system is:
[tex]\[
(x, y, z) = (-8, -1, -2)
\][/tex]
