Question

solve the following system of equations for all three variables.
$-7x - 5y + 8z = -9$
$-7x + y + 2z = -3$
$7x + 4y - 5z = -4$

Explanation:

Step1: Label the equations

Let:

$$\begin{align*} -7x - 5y + 8z &= -9 \quad \text{(1)} \\ -7x + y + 2z &= -3 \quad \text{(2)} \\ 7x + 4y - 5z &= -4 \quad \text{(3)} \end{align*}$$

Step2: Eliminate $x$ with (1)-(2)

Subtract equation (2) from (1) to remove $x$:

$$ (-7x -5y +8z) - (-7x + y +2z) = -9 - (-3) $$

$$ -6y + 6z = -6 $$

Simplify by dividing by 6:

$$ -y + z = -1 \quad \text{(4)} $$

Step3: Eliminate $x$ with (2)+(3)

Add equations (2) and (3) to remove $x$:

$$ (-7x + y +2z) + (7x +4y -5z) = -3 + (-4) $$

$$ 5y - 3z = -7 \quad \text{(5)} $$

Step4: Solve for $y$ using (4)

Rearrange equation (4) to express $z$:

$$ z = y - 1 \quad \text{(6)} $$

Step5: Substitute (6) into (5)

Plug $z = y-1$ into equation (5):

$$ 5y - 3(y-1) = -7 $$

$$ 5y -3y +3 = -7 $$

$$ 2y = -10 $$

$$ y = -5 $$

Step6: Find $z$ using (6)

Substitute $y=-5$ into equation (6):

$$ z = -5 -1 = -6 $$

Step7: Find $x$ using (2)

Substitute $y=-5, z=-6$ into equation (2):

$$ -7x + (-5) + 2(-6) = -3 $$

$$ -7x -5 -12 = -3 $$

$$ -7x -17 = -3 $$

$$ -7x = 14 $$

$$ x = -2 $$

Answer:

$x=-2$, $y=-5$, $z=-6$