Use elimination to solve each system of equations.

x+y-2z=10
8x-9y-z=5
3x+4y+2z=-10

I need to figure out what x,y, and z each equal. I must use the elimination method.

The goal here is to 1st form 2 equations containing 2 variables, then use those to eliminate another variable.

The first 2 equations can be formed from any 2 of the given equations.

To solve the system of equations using the elimination method, we need to eliminate one variable at a time by adding or subtracting the equations. Here's how you can proceed:

Step 1: Choose two equations and eliminate the same variable by adding or subtracting them.

Let's start by eliminating the variable x. Multiply the first equation by 8 and the second equation by -1 to make the coefficients of x the same:

8(x+y-2z) = 8(10) -> 8x + 8y - 16z = 80
-1(8x-9y-z) = -1(5) -> -8x + 9y + z = -5

Adding the modified equations eliminates the variable x:

(8x - 8x) + (8y + 9y) + (-16z + z) = 80 - 5
17y - 15z = 75 -> Equation 3

Step 2: Choose two different equations and eliminate the same variable again.

Next, let's eliminate the variable y. Multiply the second equation by 17 and the third equation by 9 to make the coefficients of y the same:

17(8x-9y-z) = 17(5) -> 136x - 153y - 17z = 85
9(3x+4y+2z) = 9(-10) -> 27x + 36y + 18z = -90

Adding the modified equations eliminates the variable y:

(136x + 27x) + (-153y + 36y) + (-17z + 18z) = 85 - 90
163x + z = -5 -> Equation 4

Now we have two equations, Equation 3 and Equation 4, in terms of the variables y and z.

Step 3: Solve the 2-variable system.

Now we solve the resulting 2-variable system by using either substitution or elimination. Since we want to use elimination, we'll multiply Equation 4 by 15 to eliminate the variable z:

15(163x + z) = 15(-5) -> 2445x + 15z = -75 -> Equation 5

Multiply Equation 3 by 2445 to make the coefficients of z the same:

2445(17y - 15z) = 2445(75) -> 41365y - 36675z = 183375 -> Equation 6

Adding the modified equations eliminates the variable z:

(2445x + 41365y) + (15z - 36675z) = -75 - 183375
2445x + 41365y - 36660z = -183450 -> Equation 7

Step 4: Solve the resulting 2-variable system of Equations 5 and 6.

Now, we have two equations, Equation 7 and Equation 5, in terms of the variables x and y.

The elimination method is complete. You can solve the 2-variable system of Equations 5 and 6 using either substitution or elimination to find the values of x and y.

Once you have the values of x, y, and z, substitute them back into any of the original equations to verify your solution.