4.2 Solving Systems of Equations in T1hree Variables
Equations and Unknowns
One equation and one unknown can be solved.
Two equations and two unknowns can have a solution.
Three equations and three unknowns can be solved uniquely.
In general, n equations are required to find a solution for n unknowns, except when there is no solution or an
infinite number of solutions.
Determine If a Value Is a Solution
Similar to two equations/unknowns, just replace the x, y, and z values in the equations and see if it satisfies all
three equations.
[08] Does (2,-2,1) satisfy the system
3x + 2y + z = 3
2x - 3y - 2z = 8
-2x + 4y + 3z = -9
yes
Understand the Types of Solutions for 3 variables and 3 Equations
An equation in two variables is a straight line in two dimensions.
An equation in three variables is a plane in three dimensions.
See the illustrations on page 213
Solve a System of Three Equations by Elimination
1. Write all equations in form Ax + By + Cz = D
2. Eliminate one variable from 2 equations using the elimination method
3. Eliminate the same variable using the 3rd equation using the elimination method
4. Use above equations solving the 2X2 system one to solve one of variables
5. Use the 2X2 system to get 2nd variable
6. Plug the 2 values into an equation with the 3rd variable
7. Check the solution
Examples
Solve
Solve:
[14]
x + y + z = 2
3x + y - 2z = - 2
2x - 2y + 3z = 15
(1,-2,3)
[20]
3x - 2y + z = 5
4x - 5y + 2z = 7
9x - 6y + 3z = 7
No solution
[22]
-8x + 4y + 6z = -18
2x - y - 1.5z = 4.5
4x - 2y - 3z = 9
(infinite # solutions)