1. Three Variables
We extend the general form of a linear equation to make room for a third variable.
Ax + By + Cz = D
where A, B, C, D are some real numbers.
Solutions to systems of equations with three variables should be written as an ordered triple. Looks like (x, y, z).

2. Planes A linear equation in three variables DOES NOT form a line.
It forms what we call a plane. It’s a flat surface like a piece of paper, or the chalkboard.
A system of three equations in three variables is where the planes intersect (just like solutions yesterday were where the lines crossed).

3. Solving Three Equations in Three Variables
Pick any two equations and eliminate a variable. (same way we did yesterday).
Pick a different pair of equations and eliminate the same variable.
Solve the resulting pair of equations in two variables.
To find the value of the third variable, substitute the values of the two variables found in Step 3 into any equation containing all three variables and solve the equation.
Check the solution in all three of the original equations.

4. Solve the system:
x + y + z = 4
x + y - z = 6
2x – 3y + z = -1

5. Solve the system:
2x + y + 4z = 16
x + 2y + 2z = 11
3x – 3y – 2z = -9

6. Solve the system:
2x + y - z = 1
x + 2y + 2z = 2
4x + 5y + 3z =3

7. Applications
Integer Problem: The sum of three integers is 18. The third integer is four times the second, and the second integer is 6 more than the first. Find the integers.

8. Applications
An artist makes three types of ceramic statues at a monthly cost of $650 for 180 statues. The manufacturing cost for the three types are $5, $4, and $3. If the statues sell for $20, $12, and $9, respectively, how many of each type should be made to produce $2,100 in monthly revenue?

9. Homework
P.183/ odds, evens, 28