When we talk about linear equations, mostly deal with linear equations with one variable, 2 variables and 3 variables. Linear equation which one variable which is a graph on a line, Equations with two variables whose graph plotted on a plane, also two variables require two equations to have a result for the unknowns. For three equations with three variables to have a solution, the planes must intersect in a single point. Like systems with two equations with two variables, you may need to add the opposite of one of the equations or even multiply one of the equations before adding a variable to remove one of the variables. To make it easier, rewrite the equations to have the same format. All variables are located on the left side of the equals sign and only a constant number on the right side. Systems of three equations in three variables are useful for solving many different types of real - world problems. Systems with three equations in three variables are useful for solving many different types of real problems.

Standard Form

Linear equations with 3 variables can be written as:

ax + bx + cz + d = 0

Where a, b, c and d are real numbers and a, b, c are non zero.

Solving System of Linear Equations with 3 Variables

Systems of equations in three variables that are dependent could result from three identical planes. Solving equations in three variables is similar as used to solve systems of two equations in two variables. The combination of equations is a good way to solve a system of equations, including systems with three equations and three variables. Replace known variables with one of the original equations and look for the missing variable.
Exchange the equation with two equations so that the two equations match three variables. As with equations in two variables, we also get an inconsistent system of equations in three variables, which means that system does not have any solution. 

Steps to Solve Linear Equations with 3 variables


1. Choose an equation pair of equations and solve for a variable.

2. Choose another equation pair and solve for the same variable.

3. You have created a system of two equations in two unknowns. Which will be similar to linear equations with two variables.

4. Use any of the methods to solve pair of equations with 2 variables.

5. Re-substitute the values and get results for all the variables.

6. Write the answer to a system of three equations in three variables, which is also called an ordered triple

3 Variables System of Equations Problems

Let us solve an example using above listed steps to find the ordered triples.

Example
: Solve three variable equations are

2a - b + 3c = 12

a + b - 2c = -3

2a - b + c = 6

Find the value of variables?

Solution:

Number for all equations,

2a - b + 3c = 12 ------------------ (1)

a + b - 2c = -3 ------------------ (2)

2a - b + c = 6 ------------------ (3)

Add the equations (1) and (2)

2a - b + 3c = 12

a + b - 2c = -3

After solving this so, we get

3a + c = 9 -------------- (4)

Now, add the equations (2) and (3)

a + b - 2c = -3 

2a - b + c = 6

After solving this , we get

3a - c = 3 -------------- (5)

Now, add equation (4) and (5)

3a + c = 9

3a - c = 3

After solving this we get,

6a = 12

a = 2

Insert a = 2 in equation (4)

3a + c = 9  
3(2) + c = 9

6 + c = 9

c = 9 - 6

c = 3

Insert c = 3 and a = 2 in any one equation among first three,

Here, consider a + b - 2c = -3 ......(2)

2 + b - 2 (3) = -3

2 + b - 6 = - 3

b - 4 = -3

b = 1

Thus, the solutions are a = 2, b = 1 and c = 3.