solve the system of equations with three variables

Question

7x + y + 5z = 27

4x + 3y + 5z = 21

6x + y + 2z =9

Answer 1

Eq 1: 7x + y + 5z = 27

Eq 2: 4x + 3y + 5z = 21

Eq 3: 6x + y + 2z = 9

Note that there are a multitude of ways to solve these but I'm going to choose the elimination method.

First choose the variable you wish to eliminate first. I'll choose "y"

Multiply Eq 1 by -3 and add it to Eq 2:

-21x - 3y - 15z = -81

4x + 3y + 5z = 21

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-17x - 10z = -60

Make this Eq 4

Eq 4: -17x - 10z = -60

Multiply E3 3 by -3 and add to Eq 2:

-18x - 3y - 6z = -27

4x + 3y + 5z = 21

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-14x - z = -6

Make this Eq 5

Eq 5: -14x - z = -6

Using Eq 4 and Eq5, choose a second variable to eliminate. I'll choose "z".

Multiply Eq 5 by -10 and add to Eq 4:

140x + 10z = 60

-17x - 10z = -60

-------------------------

123x = 0

x = 0

Plug this into Eq 5 to solve for "z":

-14(0) - z = -6

z = 6

Plug x = 0 and z = 6 into Eq 3 to solve for "y":

6(0) + y + 2(6) = 9

y + 12 = 9

y = -3