The system of Linear equations is of 3 types: one variable, two variables, and three variables. Three variable linear equation problems can be solved by a primary method known as the Gaussian elimination method which is named after a great mathematician of Germany, Karl Friedrich Gauss. 

The main aim of the system of the linear equation of three variables is to solve and eliminate one variable at a particular time which will allow for substituting its value in the other equations so that the equation can be solved and the solution could be found. For example, if we find the value of the third equation having ‘z’ then we can easily find out and solve the solution having ‘x’ and ‘y’. 

Solution Set:

The system of the linear equation also involves a solution set which is generally referred to in three variable linear equations. The solution set is also known as an ordered triple. Ordered triple here refers to the point at which the intersection of 3 planes happens in space. For example – imagine a corner in a room rectangle in shape, that corner has 3 planes which are two walls that are joining and the floor or the roof. Where 2 walls and a floor or roof meet, that point always represents the crossing of 3 planes. As we know the system should be in upper triangular form so to maintain that we can go through the following steps:

• Changing the sequence of any of the 2 equations out of 3. 
• Multiplying both the sides of the equation by a constant nonzero.
• Adding a multiple of nonzero of 1 equation to another equation. 

How to solve linear equations in three variables:

The steps for solving the system of linear equations in three variables are:

• Firstly, take 2 equations and then solve and find out 1 variable.
• Take another 2 equations and put the value of 1 variable and solve the equation for that variable.
• Now we are left with 2 equations and 2 variables that are not known, and then we have to solve both equations.
• Then the values of the variables that we have got, and then we should put the value of the variable in the original equation and then evaluate and find out the variables which are not known. 

Practical Examples:

Practical examples of the system of linear equation are:

1. Evaluate the following equation given below:

x – 2y + 3z = 9 

-x + 3y – z = -6

2x – 5y + 5z = 17

Solution:

By adding the equation 1 and 2 we can eliminate x so 

x – 2y + 3z +(-x +3y – z) = 9 –(-6) 

Here comes the 4th equation:

y + 2z = 3. 

Now by multiplying the 1st equation by 1 and 3rd by -2 we get,

-2x + 4y -6z = 18 + (2x – 5y + 5z = 17) 

-y – z = -1. (This is the 5th equation).

On solving the 4th and 5thequations we get, 

y + 2z = 3 

-y – z = -1 

z = 2 (this is the 6th equation) 

Now putting the value of z in the 4th equation, we get 

y + 2z = 3 

y + 2*2 = 3 

y = -1.

Now, putting the value of x and y in any of the 3 equation will give us the value of x 

Let’s take example of equation 1 

x – 2y +3z = 9

x – 2*-1 + 3*2 = 9 

x = 1. 

So the values of x, y, and z are 1, -1, and 2. 

Conclusion:

As we have come to know that the system of linear equations of 3 types and the most confusing and hardest type is a linear equation with 3 variables we have come to know about how to solve the linear equation having 3 variables very easily without any confusion. In solving this, the solution set is very important as it creates a triangular shape form in the 3 variables of a linear equation. It helps in getting to the point where the intersection of 3 planes occurs.