ALGEBRA II – WORKSHEET
CHAP 4 : SOLVING SIMULTANEOUS EQUATIONS WITH THREE VARIABLES
I.
II.
III.
x+y+z=4
x – 2y – z = 1
2x – y – 2z = -1
To solve this problem, let’s begin by eliminating ‘z’ from
equations I, II
x+y+z=4
x – 2y – z = 1
IV.
2x – y = 5
2x + 2y +2z = 8
2x – y -2z = -1
V.
Add the two equations together
4x + y
Now, let’s eliminate ‘z’ from equations I,III
Start by multiplying both sides of equation I
by the number 2, so that ‘2z’ from equation I
and III will cancel out, when added together.
=7
Now we have two equations, with two unknown
variables (see equations IV and V)
Let’s now eliminate ‘y’ from both of these
equations
VI.
2x – y = 5
4x + y = 7
This should be easy, since ‘-y’ and ‘y’
will cancel out when added together
6x = 12
Therefore we know x = 2
2(2) – y = 5
Go up to equation IV and substitute for x
4–y=5
Therefore we know y = -1
(2) + (-1) + z = 4
Go up to equation I and substitute for x and y
2–1+z=4
Which finally gives us the value of our last
variable ‘z,’ such z = 3
The solution to the given three equations (a point they have in common) is (2, -1, 3)
This ‘point’ actually represents a point in a three dimensional space, in which the lines intersect.
Document1
G.Martinson