Solving Systems of Linear
equations with 3 Variables
To solve for three variables, we need
a system of three independent
equations.
Independent, Dependent and
Inconsistent Equations
• Systems of three equations in three variables
follow the same characteristics of systems of
equations in two variables
– Independent equations have one solution
– Dependent equations have an infinite number of
solutions
– Inconsistent equations have no solution
• Solving a system of equations in three variables
involves a few more steps, but is essentially the
same process as for systems of two equations in
two variables.
Steps to solving a system of equations
in 3 Variables by Elimination
• Ensure that the equations are in standard form:
Ax + By + Cz = D
• Eliminate one of the variables using two of the three equations.
Result will be a new equation with two variables.
• Eliminate the same variable using another set of two equations.
Result will be a second equation in two variables.
• Solve the new system of two equations.
• Using the solution for the two variables, substitute the values
into one of the original equations to solve for the third variable.
• Check the solution set in the remaining two original equations.
Example system of three equations
Use elimination to solve the following system of
equations:
x + 2y - 3z = -2
2x - 2y + z = 7
x + y + 2z = -4
First, we need to ensure that all equations are in standard
form, i.e. all variables are on the left side of the equation.
Note that equation #3 is not in standard form. We need to
get the y and z terms to the left side of the equation.
Use elimination to solve the following system of
equations:
-x + y + 2z = 7
2x + 3y + z = 1
-3x - 4y + z = 4
Use elimination to solve the following system of
equations:
-x + y + 2z = 7
2x + 3y + z = 1
-3x - 4y + z = 4
Solution (-2, 1, 2)
Steps to solving a system of equations
in 3 Variables by Cramer’s Rule
• To apply Cramer’s rule to 3 x 3 systems, you need to
find the determinant of a 3 x 3 matrix.
• To find the determinant:
1st step – Rewrite the first two columns at
the right side of the determinant.
2nd step - Add the products of each down
diagonal.
3rd step – Subtract the sum of the up diagonal.
• Cramer’s Rule can be expanded to cover 3 x 3 systems.
Use Cramer’s Rule to solve the following system
of equations:
4x - 2y + z = -6
3x + 3y + 5z = -8
2x - 4y - 3z = 2
Use Cramer’s Rule to solve the following system
of equations:
4x - 2y + z = -6
3x + 3y + 5z = -8
2x - 4y - 3z = 2
Solution (1, 3, -4)