I need help with ordered triple points again I'm still confused how to do it.

Question

2x - 3y - 4z = -21, -4x + 2y - 3z = -14, -3x - 4y + 2z = -10  these are the equaltions

Answer 1

If you order the equations, then it becomes a simplified process that you can do in any situation. So first, let's order them:

 

Equation 1: 2x - 3y - 4z = -21

 

Equation 2: -4x + 2y - 3z = -14

 

Equation 3: -3x - 4y + 2z = -10

 

 

Next, the steps of the process are as follows:

 

Using one of the equations, solve for any variable. It will be in terms of the other variables, but don't worry about it just yet.

 

Next, plug that variable into a different equation, and solve for a different variable (which will now be in terms of the last unsolved variable).

 

Last, plug the first variable you solved for into the equation you didn't use yet, then replace one of the two other variables with the second variable you solved for, and simplify.

 

After those steps are completed, you will have the exact value of one variable, which then gives you the exact value of the second variable, and so on.

 

 

 

So let's try it out with these three equations. To keep things simple, let's go through the equations in order, and solve for the variables in alphabetic order.

 

 

Equation 1 says 2x - 3y - 4z = -21, and if we add 3y and 4z to the equation we get 2x = 3y + 4z - 21. Dividing by two, we then get x = 1.5y + 2z - 10.5.

 

Plugging that into equation two, we get -4x + 2y - 3z = -4(1.5y + 2z - 10.5) + 2y - 3z = -14. After simplifying, this becomes -4y - 11z = -56.

If we add 4y and 56 to the equation we get 56 - 11z = 4y. Dividing by four, we then get 14 - 2.75z = y (or y = 14 - 2.75z).

 

Now, we plug all that into equation 3, which says -3x - 4y + 2z = -10. Subsituting x first, we get -3(1.5y + 2z - 10.5) - 4y + 2z = -8.5y -4z + 31.5 = -10.

Subtracting 31.5 from the equation, we then get -8.5y -4z = -41.5.

 

Now we substitute for y, which gives us -8.5(14 - 2.75z) - 4z = -119 + 23.375z - 4z = -41.5, which we can simplify to find 19.375z = 77.5.

Dividing both sides by 19.375, we get z = 4. And since y = 14 - 2.75z, y = 14 - 2.75(4) = 14 - 11 = 3.

Now we just solve for x, and we found x = 1.5y + 2z - 10.5, so x = 1.5(3) + 2(4) - 10.5 = 2.

 

 

So we found x = 2, y = 3, and z = 4, so the answer is (2, 3, 4).

 

 

This process can be done with any number of equations and variables, so long as the number of variables doesn't exceed the number of equations.