1. pencil, red pen, highlighter, notebook, calculator
4/15/13
Have out:
Bellwork:
1) Find the solution to each system of equations. a)
2x + 3y = –17 b)
3x + 2y = 11
y = x – 4
4x + 3y = 14
2) Explain how your solution relates to the graph of each system.
total:

2. 1. Find the solution to each system of equations.
3(3x + 2y = 11)
3x + 2y = 11 a)
2x + 3y = –17 b)
–2(4x + 3y = 14)
4x + 3y = 14
y = x – 4
9x + 6y = 33 +1 +1
2x + 3(x – 4) = –17 +
–8x – 6y = –28
2x + 3x – 12 = –17
x = 5 +1
5x – 12 = –17
y = x – 4
3(5) + 2y = 11 +1
y = (–1) – 4 +1
5x = –5
15 + 2y = 11
y = –5 +1
– –15 +1
x = –1
2y = –4
(–1, –5)
(5, –2) +1 +1
y = –2 +1
2) Explain how your solution relates to the graph of each system.
The solutions represent the point of intersection for each system.
total: +2

3. Today we are going to solve systems of equations in 3 variables.
Any equation that can be written in the form _______________ is a linear equation in 3 variables (where a, b, c, and d are constants and a, b, c are not all zero).
ax + by + cz = d
The solution to such an equation is an __________________.
ordered triple (x, y, z)
The graph of a linear equation in 3 variables is a _____.
plane z x y
plane

4. Given a systems of 3 linear equations in 3 variables, there can be one solution, infinitely many solutions, or no solution.
One Solution
Infinitely Many Solutions
 planes intersect in exactly one point
 planes intersect in a line
 planes intersect in the same plane
No Solution
 planes have no point in common

5. Solving a system of equations in 3 variables is similar to solving a system of equations in 2 variables using substitution and/or elimination methods.
Example #1: Solve the system of equations. A
x + y + z = 6
Sometimes it helps to number or letter each equation. B
2x – y + 3z = 9
–x + 2y + 2z = 9 C
Dealing with 3 variables can be complicated, so let’s try to eliminate one of the variables right away.
Which two equations could we pick to eliminate a variable?
A and C

6. Example #1: Solve the system of equations.
x + y + z = 6 B
2x – y + 3z = 9
–x + 2y + 2z = 9 C
x + y + z = 6 A
Begin by adding A and C together.  This will eliminate an x–term.
–x + 2y + 2z = 9 C
Label the new equation D.
3y + 3z = 15 D
Let’s pair up two more equations to eliminate x. Which ones could we pick?
A and B
Multiply A by –2 to help eliminate x.
–2x – 2y – 2z = –12
–2(x + y + z) = –2(6) A
x + y + z = 6
2x – y + 3z = 9
2x – y + 3z = 9 B
– 3y + z = –3 E
Label the new equation E.

7. Example #1: Solve the system of equations.
x + y + z = 6 B
2x – y + 3z = 9
–x + 2y + 2z = 9 C
Now we have a couple 2-variable linear equations, D and E.
Solve for y and z.
3y + 3z = 15 D
– 3y + (3) = –3
– 3y + z = –3 E
– 3y = –6
4z = 12
y = 2
z = 3
Pick one of the original 3–variable equations to solve for x.
x + y + z = 6 A
The solution is the ordered triple
x + (2) + (3) = 6
(1, 2, 3)
x + 5 = 6
x = 1

8. Example #2: Solve the system of equations.
x + 2y + z = 10
It helps to number or letter each equation. B
2x – y + 3z = –5
2x – 3y – 5z = 27 C
Pick any two equations to eliminate a variable.
Let’s eliminate y for the fun of it. 
Choose B and C B
–3(2x – y + 3z) = –3(–5)
2x – y + 3z = –5
–6x + 3y – 9z = 15
2x – 3y – 5z = 27 C
2x – 3y – 5z = 27
– 4x – 14z = 42 D
Label the new equation D.
Pick another pair of equations to eliminate the y–term.
A and B
x + 2y + z = 10 A
x + 2y + z = 10
2(2x – y + 3z) = 2(–5)
2x – y + 3z = –5
4x – 2y + 6z = –10 B
5x z = 0 E
Label the new equation E.

9. Now we have a couple 2-variable linear equations.
Solve for x and z.
– 4x – 14z = 42 D
– 4x – 14z = 42
5x + 7z = 0
2(5x + 7z) = 2(0)
5x + 7z = 0 E
10x + 14z = 0
5(7) + 7z = 0
35 + 7z = 0
6x = 42
7z = –35
x = 7
z = –5
Pick one of the original 3–variable equations to solve for y.
x + 2y + z = 10
(7) + 2y + (–5) = 10
The solution is the ordered triple
2 + 2y = 10
(7, 4, –5)
2y = 8
y = 4

10. Summary:
1. Simplify and put all three equations in the form ___________ if needed.
Ax+By+Cz = D
variables
2. Choose to _________ any one of the _________ from any ______________.
eliminate
pair of equations
3. Eliminate the ______ variable chosen in step 2 from any ___________________, creating a system of two equations and 2 unknowns.
SAME
other pair of equations
system of 2 equations
4. Solve the remaining __________________ (with _ unknowns) found in step 2 and 3. 2
third variable
5. Solve for the ___________. Write the answer as an __________________.
ordered triple (x, y, z).
6. Check.

11. Complete the worksheets
and the CST practice.

12. References & Resources