Systems of Equations with 3 Variables

elimination

determine two different combinations of the three equations to eliminate the SAME variable from each use two equations with two variables and eliminate to solve for one of the variables plug back into equations to find other two variables

no solution

false equation when substituting/eliminating

substitution

isolate one variable and plug in to two other equations to get two equations with two variables use elimination to solve for one of the variables plug into equations to solve for other two variables

how to write infinite solutions

let y=t plug t into the equation solve for other variable plug into one of the original equations to get other variable write three variable terms as ordered triple

no solution definition

no point lies in all three planes

infinite solutions

one true equation and one line equation

if not all three variables are in every equation

pick two variables solve for one plug into equation with only one of the selected variables use other equation to continue on to elimination, etc.

infinite number of solutions

the planes intersect at all the points along a common line

one solution definition

the planes intersect at one common point

ordered triple

three variables' solutions, written in alphabetical order