Get A+ with YaClass!
Register now to understand any school subject with ease and get great results on your exams!

### Theory:

An equation in which three variables $$x$$, $$y$$ and $$z$$ are of the first degree, then the equation is said to be a linear equation in three variables.

The general form of linear equation in three variables can be written as:

$$ax + by + cz + d = 0$$

Here, atleast one of $$a$$, $$b$$, $$c$$ and $$d$$ are non - zero, and they are real numbers and $$x$$ and $$y$$ are variables.
Graphical representation
A linear equation in two variables is of the form $$ax + by + c = 0$$ represents a straight line.

A linear equation in three variables is of the form $$ax + by + cz + d = 0$$ is a plane.

Solution of the system of equations
The general form of a system of linear equation in three variables is given by:

$$a_1x + b_1y + c_1z + d_1 = 0$$

$$a_2x + b_2y + c_2z + d_2 = 0$$

$$a_3x + b_3y + c_3z + d_3 = 0$$

Exactly one solution: Three equations meet at a common point in a plane, then the system of equations have only one solution.

Infinitely many solutions: If three equations lie on the same plane, then the system of equations has infinitely many solutions.

No solution: Three equations do not have any common point.

Steps to solve the system of linear equations in three variables
Step 1: Consider any $$2$$ equations from $$3$$ equations. Multiply the equations with suitable values such that any one of the variables gets cancelled, leaving the variables either $$x$$ and $$y$$ or $$x$$ and $$z$$ or $$y$$ and $$z$$.

Step 2: Again, consider any $$2$$ equations from $$3$$ equations and eliminate the same variable which was eliminated in the previous pair.

Step 3: Now, we have $$2$$ equations with two variables.

Step 4: Solve these $$2$$ equations using any method like substitution or elimination or cross multiplication method.

Step 5: Substitute the value of these $$2$$ variables in any of the given equations and determine the value of the third variable.
Important!
1. In any of the steps, if we get a false equation like $$0 = 1$$, then the system is inconsistent and has no solution.

2. If we get an equation like $$0 = 0$$, then the system is consistent and has infinitely many solutions.