Lining the letters up vertically as below, makes it easier
to see what's what:
3x + 3y = 3
-3x - 2z = -11
y + z = -1
The idea is to reduce this 3x3 system to a 2x2 system.
This is done by using the three equations to obtain
only two equations in only two unknowns.
1. We observe that x is already eliminated from the 3rd
equation
and
2. We observe that x is easily eliminated from the 1st
and 2nd equations just by adding them term by term
just as they are, since the coefficients of x in them
are equal in absolute value but opposite in sign.
So we add them term by term:
3x + 3y = 3
-3x - 2z = -11
-------------------
3y - 2z = -8
3. Next we solve the system
3y - 2z = -8
y + z = -1
We can eliminate z if we multiply the 2nd equation (which
was the 3rd original equation) by 2 and adding them term by term:
3y - 2z = -8
2y + 2z = -2
------------
5y = -10
y = -2
4. Then we substitute -2 for y in
y + z = -1
-2 + z = -1
z = 1
5. Then finally we substitute -2 for y in
the first original equation
3x + 3y = 3
3x + 3(-2) = 3
3x - 6 = 3
3x = 9
x = 3
Solution (x,y,z) = (3,-2,1)
Edwin
