Solve systems with three variables by using substitution and elimination.

 Systems of equations with 3 variables can be represented as graphs in 3 dimensions.
 The graph of the equation Ax + By + Cz = D is a plane
 The solutions of a three-variable system is the intersection of the planes.

 Solve the system.
 We can pair the equations to eliminate z.
 Now we have 2 new equations with only 2 variables so we can use elimination to solve for y.
 Substitute y into equation 4 or 5 to solve for x.
 Substitute x and y into one of the original equations to solve for z. The solution is the ordered triple (x, y, z)
 (3, 3, 1)


 (1, -4, 3)
Use elimination

 Solve the system.
 x in equation 2 looks easiest to isolate
 x = 9 – 5y
 Substitute 9 – 5y for x in equations 1 and 3
 Now we have 2 equations in 2 variables
 Solve using substitution or elimination
 Eliminating z looks easiest in this case to solve for y.
 Substitute y into equation 4 or 5.
 Substitute known values in one of the original equations to find x.
 (4, 1, 6)

 You manage a clothing store and budget $6000 to restock 200 shirts.
You buy t-shirts for $12 each, polos for $24 each, and rugby shirts for
$36 each. You want twice as many rugby shirts as polos. How many of each shirt should you buy?
 Define variables: x = t-shirts, y = polos, z = rugby shirts
 Write equations:

 x + y + z = 200 (you want a total of 200 shirts)
12x+24y+36z = 6000 (you have a $6000 budget)
 z = 2y (twice as many rugby shirts as polos)
 We can write equation 2 as a simpler equivalent equation: x + 2y +3z = 500 (divide by 12)
 Use substitution
 20 t-shirts, 60 polos, 120 rugby shirts

 Odds p.171 #9-13,21-25,31-35