3.6 Systems with Three Variables
Great ability develops and reveals itself
increasingly with every new assignment.
Review!!!
slope
yintercept
Different
Doesn’t matter
Same
Same
Same
Different
Now in 3D!!!
No Solution because no point
lies in all 3 planes
One Solution because only 1
point lies in all 3 planes
Infinitely Many Solutions because the
planes intersect at all of the points along
a common line
Solving by Elimination
2 x  y  z  5

Ex1) 3x  y  2 z  1
x  y  z  0

1) Pair equations to eliminate one
variable
2) Write the two new equations
as a system and solve for the
remaining 2 variables
3) Substitute those values into
one of the original equations
and solve for the third variable
4) CHECK!! 
Solving by Elimination
 x  4 y  5 z  7

Ex2) 3 x  2 y  3z  7
2 x  y  5 z  8

Solving by Substitution
x  3y  z  6

Ex3) 2 x  5 y  z  2
 x  y  2 z  7

1) Choose one equation to solve for
one of its variables
2) Substitute the expression for that
variable into each of the other 2
equations
3) Write the 2 new equations as a
system and solve for the two
remaining variables
4) Substitute your 2 values into an
original equation and solve for the
third
5) CHECK!! 
3.6 Systems with Three Variables
HW: #1, 3, 5, 11, 13, 20, 21
Great ability develops and reveals itself
increasingly with every new assignment.