How to Solve Simultaneous Equation with 3 Variables

by Wonder Boy · Published August 12, 2019 · Updated April 15, 2020

Let’s take simultaneous equation with 3 variables.

$a_1x + a_2y + a_3z = a_4$ ……………(1)
$b_1x + b_2y + b_3z = b_4$ …………..(2)
$c_1x + c_2y + c_3z = c_4$ …………….(3)

To solve simultaneous equation with 3 variables
Step 1: Take 2 equations, say (1) & (2) to remove any one variable, say z. Mark this equations as (4).
Step 2: Take another set of 2 equations, say (2) & (3) to remove the same variable which was removed in step 1, say z. Mark this equations as (5).
Step 3: Now (4) & (5) are simultaneous equation in 2 variables x & y.
Step 4: Solve equation (4) & (5) to get the values of x & y.
Step 5: Put value of x & y in equation (1) or (2) or (3), and solve to get value of z. Finally we have the solution of all the three variables x,y & z.

Let’s take an example of simultaneous equation with 3 variables.
x + y – z = 4 ……………(1)
x − 2y + 3z = -6 ……………(2)
2x + 3y + z = 7 ……………(3)

Adding equation (2) with 3 X equation (1) to eliminate the variable z, we get:
x-2y+3z + 3x+3y-3z = -6 +12
4x+y = 6 …………..(4)

Adding equation (3) & equation (1) to eliminate the variable z, we get:
x+y-z + 2x+3y+z = 11
3x + 4y = 11 ……………..(5)

Now, Subtracting equation (5) from 4 X equation (4), we get
3x+4y – 16x-4y = 11-24
-13x = -13
x = 1
From equation (4), y = 6- 4*1 = 2

From equation (1), z = x+y-4 = 1+2-4 = -1

Solution of the equation is (1,2,-1) or (x=1, y =2, z=-1)