Simultaneous Equations | Three Unknowns

Solving simultaneous equations can be done with many methods. If you understand the underlying concept of elimination, the process will be simple and fun to use.

In this post, the Elimination method to solve 3 unknowns is presented. In a previous post of this website, the steps to solve simultaneous equations having 2 unknowns using Elimination method can be reviewed by clicking this link.

The concept to solving simultaneous equations with 3 unknowns is the same, except the steps are more, since more equations and unknowns are present. Logical, right?

Elimination means the removal of variables or unknown in the process of solving. Study the approach or concept as it serves a good foundation which can be applied to any quantity of unknowns.

The strategy used in the Elimination method of solving is to choose a random unknown to be removed. This reduces the 3-unknown type of question to a simpler 2-unknown type question that can be easily dealt with. The details are presented below.

Letus begin with an example.

x + 2y + 3z = 10 --- (A)
2x + y + z = 9 --- (B)
3x + y + 2z = 13 --- (C)

Step 1:
Look at the 3 equations and decide which unknown is easy to remove. From equation (B) and (C), we can see that the unknown "y" can be easily eliminated by just subtracting the 2 mentioned equations.

(C) - (B): x + 0 + z = 4 --- (D) ==> unknown "y" disappeared !

Step 2:
Find another pair of equations with the purpose of removing the same unknown in step 1.
In this example, there is no 2 other combinations that allow us to remove the same unknown just by performing adding or subtracting. We need to do something first before we can move on in step 2.

Let's target the use of equation (A) and (B) to remove the "y" unknown. Note this is randomly selected.

We see that by multiplying equation (B) by 2, we get 4x + 2y + 2z = 18 ---(E).
The "2y" is now the same as that in equation (A).

Rewriting for simplification,

x + 2y + 3z = 10 --- (A)
4x + 2y + 2z = 18 ---(E)

(A) - (E): -3x + 0 + z = -8 --- (F)

Step 3:
Now we have obtained 2 new equations with only 2 unknowns.
This is the strategy to reduce the more complex 3-unknown question to the simpler 2-unknown question.

Rewriting the 2 new equations here:
x + 0 + Z = 4 --- (D)
-3x + 0 + z = -8 --- (F)

From here onwards, you can review the steps by clicking the link here. It teaches you how to proceed with solving 2-unknown simultaneous equation using the Elimination method.

Some common mistakes made:

The sign of the variables and numbers in the equation are not done correctly after subtraction of 2 equations. Example of this is can be seen using the above question.

If we subtract equation (F) from (D), we should get 4x + 0 = 12.

The common error is to do the x -3x instead of x -(-3x) = x + 3x. The "-" sign error !

To avoid this type of careless mistake, focus on the sign when doing subtraction of equations.
Just keep your mind clear, and constantly remind yourself of this "slip of the mind" mistake.

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