Question: User: 10x - 4y - 7z = -9
-7x + 4y + 6z = 2
-6x - 2y + 5z = -3
Three equations in a system, centered on the page; position_hint = center
1. **State the problem:**
We are given a system of three linear equations:
$$10x - 4y - 7z = -9$$
$$-7x + 4y + 6z = 2$$
$$-6x - 2y + 5z = -3$$
We need to find the values of $x$, $y$, and $z$ that satisfy all three equations simultaneously.
2. **Write the system clearly:**
$$\begin{cases}
10x - 4y - 7z = -9 \\
-7x + 4y + 6z = 2 \\
-6x - 2y + 5z = -3
\end{cases}$$
3. **Use elimination or substitution to solve:**
Let's use elimination to reduce the system step-by-step.
4. **Add the first two equations to eliminate $y$:**
$$ (10x - 4y - 7z) + (-7x + 4y + 6z) = -9 + 2 $$
Simplify:
$$ (10x - 7x) + (-4y + 4y) + (-7z + 6z) = -7 $$
$$ 3x + 0y - z = -7 $$
So,
$$ 3x - z = -7 \quad \Rightarrow \quad z = 3x + 7 $$
5. **Use the third equation and substitute $z$:**
$$ -6x - 2y + 5z = -3 $$
Substitute $z = 3x + 7$:
$$ -6x - 2y + 5(3x + 7) = -3 $$
Simplify:
$$ -6x - 2y + 15x + 35 = -3 $$
$$ ( -6x + 15x ) - 2y + 35 = -3 $$
$$ 9x - 2y + 35 = -3 $$
Subtract 35 from both sides:
$$ 9x - 2y = -38 $$
6. **Use the first equation and substitute $z$:**
$$ 10x - 4y - 7z = -9 $$
Substitute $z = 3x + 7$:
$$ 10x - 4y - 7(3x + 7) = -9 $$
Simplify:
$$ 10x - 4y - 21x - 49 = -9 $$
$$ (10x - 21x) - 4y - 49 = -9 $$
$$ -11x - 4y - 49 = -9 $$
Add 49 to both sides:
$$ -11x - 4y = 40 $$
7. **Now solve the two equations with $x$ and $y$ only:**
$$ \begin{cases}
9x - 2y = -38 \\
-11x - 4y = 40
\end{cases}$$
8. **Multiply the first equation by 2 to align $y$ coefficients:**
$$ 2(9x - 2y) = 2(-38) $$
$$ 18x - 4y = -76 $$
9. **Subtract the second equation from this new equation:**
$$ (18x - 4y) - (-11x - 4y) = -76 - 40 $$
Simplify:
$$ 18x - 4y + 11x + 4y = -116 $$
$$ 29x = -116 $$
10. **Solve for $x$:**
$$ x = \frac{-116}{29} = -4 $$
11. **Substitute $x = -4$ into $9x - 2y = -38$ to find $y$:**
$$ 9(-4) - 2y = -38 $$
$$ -36 - 2y = -38 $$
Add 36 to both sides:
$$ -2y = -2 $$
Divide both sides by $\cancel{-2}$:
$$ y = \frac{-2}{\cancel{-2}} = 1 $$
12. **Substitute $x = -4$ into $z = 3x + 7$ to find $z$:**
$$ z = 3(-4) + 7 = -12 + 7 = -5 $$
**Final solution:**
$$ x = -4, \quad y = 1, \quad z = -5 $$