Question

1. 2x - 6y = 6z - 6y

x = 2z - 1
3z = 18

1. y = 2x - 6

6x = 12
x - 2y + 5z = -24

Explanation:

Step1: Solve for $z$ in the first - set of equations

Given $3z = 18$, divide both sides by 3.
$z=\frac{18}{3}=6$

Step2: Substitute $z$ into the equation for $x$

Since $x = 2z-1$ and $z = 6$, then $x=2\times6 - 1=12 - 1 = 11$.

Step3: Simplify the first equation in the first - set

Starting with $2x-6y=6z - 6y$, add $6y$ to both sides. We get $2x=6z$. Substituting $x = 11$ and $z = 6$, the left - hand side is $2\times11 = 22$ and the right - hand side is $6\times6=36$. There is an error in the first set of equations as they are inconsistent.

For the second set of equations:

Step1: Solve for $x$

Given $6x = 12$, divide both sides by 6.
$x=\frac{12}{6}=2$

Step2: Substitute $x$ into the equation for $y$

Since $y = 2x-6$ and $x = 2$, then $y=2\times2-6=4 - 6=-2$

Step3: Substitute $x$ and $y$ into the third equation

Substitute $x = 2$ and $y=-2$ into $x-2y + 5z=-24$.
We get $2-2\times(-2)+5z=-24$.
First, simplify the left - hand side: $2 + 4+5z=-24$.
$6+5z=-24$.
Subtract 6 from both sides: $5z=-24 - 6=-30$.
Divide both sides by 5: $z=\frac{-30}{5}=-6$

Answer:

For the first set of equations, they are inconsistent. For the second set of equations: $x = 2$, $y=-2$, $z=-6$