Solve the following system of equations for all three variables.

Question

x-y+z=-6

3x-y-3z = 10

x+y + z= -8

x= y= z=

Answer 1

Here you have a system of three simultaneous equations in three unknown variables x, y, z.

(1) x-y+z= -6

(2) 3x-y-3z = 10

(3) x+y + z= -8

The process is to take each pair of equations (1 and 2, or 1 and 3, or 2 and 3) and see if you can use a combination of multiplication and subtraction/addition to eliminate one variable.

In this case, I'm immediately struck by the similarity of (1) and (3), so I want to start with these. As they both have a single x, we don't need any multiplication, just subtraction will work!

So I want (1) - (3)

x - y + z = -6

-

x + y + z = -8

=

0x - 2y + 0z = 2

=> -2y = 2

=> y = -1

In this case, we get rid of 2 variables in one go, we usually won't be so lucky!

Now we know y, we can plug it into the equations. For 2 variables we only need 2 equations, so I won't bother to figure out the third.

(1) x - y + z = -6

=> x -(-1) + z = -6

=> x + 1 + z = -6

=> x + z = -7

(2) 3x - y - 3z = 10

=> 3x - (-1) - 3z = 10

=> 3x + 1 - 3z = 10

=> 3x - 3z = 9

Now our system of two simultaneous equations in two unknowns is:

(4) x + z = -7

(5) 3x - 3z = 9

You have single x in one, and 3x in the other, so we need some multiplication.

3 x equation (4) is 3x + 3z = -21

Now additiion will eliminate z:

3x + 3z = -21

+

3x - 3z = 9

=

6x + 0z = -12

=> 6x = -12

=> x = -2

Now we have the value of x, we just substitute it in either (4) or (5) to find z:

x + z = -7

=> -2 + z = -7

=> z = -5

Hence our solution is x = -2, y = -1 and z = -5.

As always, it's wise to check your answer, which we do by substituting these values in one of the original equations.

x + y + z = -8

=> (-2) + (-1) + (-5) = -8

=> -2 - 1 - 5 = -8

=> -8 = -8

oooo... that's a bingo! :-)