SOLUTION: How do you solve this system using elimination? -2x+5y-3z=7 4x-3y+2z=4 -3x-y-4z=-7

Question 1132567: How do you solve this system using elimination?
-2x+5y-3z=7
4x-3y+2z=4
-3x-y-4z=-7
Found 2 solutions by t0hierry, greenestamps:
Answer by t0hierry(194) (Show Source): You can put this solution on YOUR website!
-2x+5y-3z=7 (multiply by 2) -4x + 10y - 6z = 14
4x-3y+2z=4 4x - 3y + 2z =4
-3x-y-4z=-7
-4x + 10y - 6z = 14
4x - 3y + 2z =4 add
7y -4z = 18
y = 2
z = -1
x = 3
Answer by greenestamps(13198) (Show Source): You can put this solution on YOUR website!

The work the other tutor did to solve this problem is not at all clear; and in fact he shows the correct solution without showing all the work he did to get it.

It won't help you learn anything if I show you a complete solution. The objective of the problem is to have YOU learn how to solve it.

And, in fact, since your question is "how do you solve" the system, I will tell you how you can solve the system, and YOU can do it.

The basic idea for solving a system of three equations in three variables using elimination is to manipulate the given equations in a way that eliminates one of the variables, giving you a system of two equations in two variables.

With this system, before I start doing anything else, I would change the last equation to have positive coefficients, since that often makes the algebra easier. So we have

-2x+5y-3z=7
4x-3y+2z=4
3x+y+4z=7

We can choose to eliminate any one of the variables. Look at the coefficients in the three equations and decide which variable you think will be easier to eliminate. Because the coefficient of y in the third equation is 1, I would choose to eliminate y.

But you can choose to eliminate any one of the three variables; note that the level of difficulty of the required calculations can be very different for different choices.

To eliminate y, you can do as follows:

(1) multiply the third equation by -5 and add to the first equation; that will eliminate y between those two equations and give you an equation in x and z.
(2) multiply the third equation by 3 and add to the second equation; that will also eliminate y and give you a second equation in x and z.

Solve that system of two equations in two variables, presumably by elimination, and substitute the values you find for x and z into any one of the three original equations to solve for y.
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