# Introduction to Linear Equations in Three Variables

• Linear equations can be expanded to include three variables, primarily represented as x, y, and z.

• An example of a linear equation in three variables is: ax + by + cz = d, where a, b, c, and d are constants.

# Formulation of Linear Equations in Three Variables

• The formation of linear equations in three variables follows the same principles as for two variables.

• The difference is that there is now an additional z term that needs to be incorporated into the formation and solution of the equations.

# Solving Linear Equations in Three Variables

• The main goal when solving linear equations in three variables is to use substitution and elimination methods to find the values of x, y, and z.

• To eliminate one of the variables, it is common to add or subtract the equations.

• Once one variable is eliminated, a system of two equations will be left, which can be solved using well-known two-variable methods.

# Checking the Solution for Accuracy

• After obtaining a solution, it is important to check the values that were found for x, y, and z by substituting them back into the original equations.

• If the substituted values hold true for all of the original equations, then the solution can be considered as valid.