What is the solution to the system of equations shown below?

2x-y+3z=8
x-6y-z=0
-6x+3y-9z=24

no solutions because

2x-y+3z=8
and
-6x+3y-9z=24

reduce to

2x-y+3x = 8
2x-y+3z = -8

which are parallel planes with no intersection

To solve this system of equations, we'll use the method of elimination.

Step 1: Multiply the second equation by 2 and the third equation by -1 to eliminate the x term.
2(x - 6y - z) = 2(0) => 2x - 12y - 2z = 0
-1(-6x + 3y - 9z) = -1(24) => 6x - 3y + 9z = -24

The modified system of equations becomes:
2x - y + 3z = 8
2x - 12y - 2z = 0
6x - 3y + 9z = -24

Step 2: Add the first equation to the second equation to eliminate the x term.
(2x - y + 3z) + (2x - 12y - 2z) = 8 + 0
4x - 13y + z = 8

The new system of equations becomes:
4x - 13y + z = 8
6x - 3y + 9z = -24

Step 3: Multiply the first equation by 3 and add it to the second equation to eliminate the x term.
3(4x - 13y + z) + (6x - 3y + 9z) = 3(8) + (-24)
12x - 39y + 3z + 6x - 3y + 9z = 24 - 24
18x - 42y + 12z = 0

The new system of equations becomes:
18x - 42y + 12z = 0
6x - 3y + 9z = -24

Step 4: Multiply the second equation by -6 and add it to the first equation to eliminate the x term.
-6(6x - 3y + 9z) + (18x - 42y + 12z) = -6(-24) + 0
-36x + 18y - 54z + 18x - 42y + 12z = 144
-18x - 24z = 144

The new system of equations becomes:
-18x - 24z = 144
6x - 3y + 9z = -24

Step 5: Solve one of the equations for a variable and substitute it back into the other equation.
-18x - 24z = 144 => divide both sides by -6
3x + 4z = -24

Solve this equation for x:
3x = -4z - 24
x = (-4z - 24)/3

Substitute this value of x into the second equation:
6((-4z - 24)/3) - 3y + 9z = -24
-8z - 48 - 3y + 9z = -24
y = -8z + 24

Now we have expressions for x and y in terms of z.

Step 6: Choose a value for z and calculate the corresponding values for x and y. Let's choose z = 0 for simplicity.

x = (-4z - 24)/3 = (-4(0) - 24)/3 = -8
y = -8z + 24 = -8(0) + 24 = 24

So, one solution to the system of equations is x = -8, y = 24, and z = 0.

Overall, the solution to the system of equations is x = -8, y = 24, and z = 0.

To solve the system of equations, we can use the method of elimination or substitution. Here, I'll explain both methods, and you can choose which one you prefer.

Method 1: Elimination
Step 1: Multiply the second equation by 2 to make the x-coefficient in equation 2 equal to the x-coefficient in equation 1.
2(x-6y-z) = 2(0)
This simplifies to:
2x - 12y - 2z = 0

Step 2: Add equation 1 and the modified equation 2 to eliminate the x variable.
(2x - y + 3z) + (2x - 12y - 2z) = 8 + 0
This simplifies to:
4x - 13y + z = 8

Step 3: Multiply equation 3 by 3 to make the x-coefficient in equation 3 equal to the x-coefficient in equation 1.
3(-6x + 3y - 9z) = 3(24)
This simplifies to:
-18x + 9y - 27z = 72

Step 4: Add equation 1 and the modified equation 3 to eliminate the x variable.
(4x - 13y + z) + (-18x + 9y - 27z) = 8 + 72
This simplifies to:
-14x - 4y - 26z = 80

Now we have a new system of equations:
-14x - 4y - 26z = 80 (Equation 4)
-18x + 9y - 27z = 72 (Equation 5)

Step 5: Solve the new system of equations 4 and 5.

Using the method of elimination again, we can eliminate the y variable by multiplying equation 4 by 9 and equation 5 by 4:
9(-14x - 4y - 26z) = 9(80)
This simplifies to:
-126x - 36y - 234z = 720 (Equation 6)

4(-18x + 9y - 27z) = 4(72)
This simplifies to:
-72x + 36y - 108z = 288 (Equation 7)

Adding equations 6 and 7 eliminates the y variable:
(-126x - 36y - 234z) + (-72x + 36y - 108z) = 720 + 288
This simplifies to:
-198x - 342z = 1008 (Equation 8)

Finally, solving equations 8 gives us the values for x and z.

Method 2: Substitution
Step 1: Solve either equation 2 or equation 3 for one variable in terms of the other variables. Let's solve equation 2 for x:
x = 6y + z

Step 2: Substitute this expression for x into equation 1 and equation 3:
2(6y + z) - y + 3z = 8 (Equation 9) (substituting into equation 1)
-6(6y + z) + 3y - 9z = 24 (Equation 10) (substituting into equation 3)

Now we have two equations, 9 and 10, with only the variables y and z.

Step 3: Simplify equations 9 and 10:
12y + 2z - y + 3z = 8
This simplifies to:
11y + 5z = 8 (Equation 11)

-36y - 6z + 3y - 9z = 24
This simplifies to:
-33y - 15z = 24 (Equation 12)

Step 4: Solve the new system of equations 11 and 12.

Using method 1 (elimination), we can eliminate the y variable by multiplying equation 11 by 3 and equation 12 by 11:
3(11y + 5z) = 3(8)
This simplifies to:
33y + 15z = 24 (Equation 13)

11(-33y - 15z) = 11(24)
This simplifies to:
-363y - 165z = 264 (Equation 14)

Adding equations 13 and 14 eliminates the y variable:
(33y + 15z) + (-363y - 165z) = 24 + 264
This simplifies to:
-330y - 150z = 288 (Equation 15)

Finally, solving equations 15 gives us the values for y and z.

Once you have the values for x, y, and z, you can substitute them back into any of the original equations to verify if they satisfy the equations.