Topic 4: Solving three simultaneous linear equations
Three simultaneous equations are three equations which contain three variables e.g. x, y and z and the
three equations are satisfied by the same values of x, y and z.
Three simultaneous equations can only be solved using the algebraic method of elimination.
Example 1
Solve these simultaneous equations
5x – y + 2z = 25 …………………………(A)
3x + 2y – 3z = 16………………………….(B)
2x – y + z = 9 …………………………..(C)
Solution 1
We begin by trying to eliminate one of the variables e.g y
(A) – (C)
5x – y + 2z = 25
- 2x – y + z = 9
3x
+ z = 16 ………………………….(D)
2 x (A) + (B)
10x - 2y + 4z = 50
+ 3x + 2y - 3z = 16
13x
+ z = 66 ………………………….(E)
Now we have two simultaneous equations (D) and (E) which we can easily solve
(E) – (D)
13x + z = 66
- 3x + z = 16
10x
= 50
x = 50 ÷ 10
x=5
Substitute x = 5 into (D)
3 x 5 + z = 16
15 + z = 16
z = 16 – 15
z=1
Substitute x = 5, z = 1 into (A)
5 x 5 – y + 2 x 1 = 25
25 – y + 2 = 25
27 – y = 25
27 – 25 = y
2=y
Solution is x = 5, y = 2 and z = 1