Question

Need help solving a 3 variable equation:

-3x - 4y + 2z = 28
x + 3y - 4z = -31
x - 3 y -7z = 42

Answer 1

best thing to do is take equation 1 and equation 2 and eliminate one of the variables.
then take equation 2 and 3 and eliminate that same variable and then perform simultaneous equations.

would u like to go step by step?

Answer 2

take your last 2 equations and add them...this will eliminate the y's.

x + 3y - 4z = -31
x - 3y - 7z = 42
----------------add
2x - 11z = 11

Now take one of the equations you just used, and the equation you haven't used yet...and eliminate the y's......you have to eliminate the same variable that you did in the first set of equations.

x + 3y - 4z = -31 --- multiply by 4
-3x - 4y + 2z = 28 --- multiply by 3
-----------------
4x + 12y - 16z = - 124 (result of multiplying by 4)
-9x - 12y + 6z = 84 (result of multiplying by 3)
-----------------add
-5x - 10z = - 40

Now take your 2 resulting equations and eliminate a variable
2x - 11z = 11 --- multiply by 5
-5x - 10z = -40 --- multiply by 2
-------------
10x - 55z = 55 (result of multiplying by 5)
-10x - 20z = -80 (result of multiplying by 2)
---------------add
- 75z = - 25 -- divide both sides by -75
z = -25/-75
z = 1/3

Now that we know z, we then can substitute 1/3 in for z in one of the equations that the y has already been eliminated.
2x - 11z = 11
2x - 11(1/3) = 11
2x - 11/3 = 11
2x = 11 + 11/3
2x = 33/3 + 11/3
2x = 44/3
x = (44/3) / 2
x = 44/3 * 1/2
x = 44/6 which reduces to 22/3

Now that we know z and x, we can sub that into one of the original equations to find y.
x + 3y - 4z = -31
22/3 + 3y - 4(1/3) = -31
22/3 + 3y - 4/3 = -31 -- multiply everything by common denominator of 3 to get rid of fractions.
22 + 9y - 4 = - 93 (result of multiplying by 3)
9y + 18 = -93
9y = -93 - 18
9y = -111
y = -111/9 which is -37/3

Now we must check our answers by subbing in known terms into one of the original equations. I recommend checking because 1 mistake, if made at the beginning of the problem, can carry through to the end, resulting in the wrong answer.

x - 3y - 7z = 42 (x = 22/3, y = -37/3 and z = 1/3)
22/3 - 3(-37/3) - 7(1/3) = 42
22/3 + 37 - 7/3 = 42 --- multiply everything by 3 to get rid of fractions
22 + 111 - 7 = 126
126 = 126 (correct)

so...x = 22/3, y = -37/3 and z = 1/3

This is not really that hard, just very time consuming.