The document discusses solving systems of linear equations with three variables. The standard method is elimination, where one extracts a system of two equations with two unknowns from the original system of three equations. One solves the system of two equations and plugs the answers back into the original system to find the third variable. This process generalizes to systems with N equations and N unknowns by repeatedly extracting and solving smaller systems until reaching a system of two equations. An example problem demonstrates applying this method to find the price of hamburgers, fries, and sodas given three equations relating purchases to total costs.

1. Systems of Linear Equations With Three Variables

2. Systems of Linear Equations With Three Variables
To solve for three unknowns, we need three pieces of
numerical information about the unknowns,

3. Systems of Linear Equations With Three Variables
To solve for three unknowns, we need three pieces of
numerical information about the unknowns, i.e. three
sequations.

4. Systems of Linear Equations With Three Variables
To solve for three unknowns, we need three pieces of
numerical information about the unknowns, i.e. three
sequations. The standard method for solving systems of
linear equations is the elimination method.

5. Systems of Linear Equations With Three Variables
To solve for three unknowns, we need three pieces of
numerical information about the unknowns, i.e. three
sequations. The standard method for solving systems of
linear equations is the elimination method.
We use elimination method to extract a system of two
equations with two unknowns from the system of
three equations.

6. Systems of Linear Equations With Three Variables
To solve for three unknowns, we need three pieces of
numerical information about the unknowns, i.e. three
sequations. The standard method for solving systems of
linear equations is the elimination method.
We use elimination method to extract a system of two
equations with two unknowns from the system of
three equations. Solve the system of 2 equations and plug
the answers back to get the third answer.

7. Systems of Linear Equations With Three Variables
To solve for three unknowns, we need three pieces of
numerical information about the unknowns, i.e. three
sequations. The standard method for solving systems of
linear equations is the elimination method.
We use elimination method to extract a system of two
equations with two unknowns from the system of
three equations. Solve the system of 2 equations and plug
the answers back to get the third answer.
This is also the general method for solving a system of N
equations with N unknowns.

8. Systems of Linear Equations With Three Variables
To solve for three unknowns, we need three pieces of
numerical information about the unknowns, i.e. three
sequations. The standard method for solving systems of
linear equations is the elimination method.
We use elimination method to extract a system of two
equations with two unknowns from the system of
three equations. Solve the system of 2 equations and plug
the answers back to get the third answer.
This is also the general method for solving a system of N
equations with N unknowns. We use elimination method to
extract a system of (N – 1) equations with (N – 1) unknowns.
from the system of N equations.

9. Systems of Linear Equations With Three Variables
To solve for three unknowns, we need three pieces of
numerical information about the unknowns, i.e. three
sequations. The standard method for solving systems of
linear equations is the elimination method.
We use elimination method to extract a system of two
equations with two unknowns from the system of
three equations. Solve the system of 2 equations and plug
the answers back to get the third answer.
This is also the general method for solving a system of N
equations with N unknowns. We use elimination method to
extract a system of (N – 1) equations with (N – 1) unknowns.
from the system of N equations. Then we extract a system of
(N – 2) equations with (N – 2) unknowns from the (N – 1)
equations.

10. Systems of Linear Equations With Three Variables
To solve for three unknowns, we need three pieces of
numerical information about the unknowns, i.e. three
sequations. The standard method for solving systems of
linear equations is the elimination method.
We use elimination method to extract a system of two
equations with two unknowns from the system of
three equations. Solve the system of 2 equations and plug
the answers back to get the third answer.
This is also the general method for solving a system of N
equations with N unknowns. We use elimination method to
extract a system of (N – 1) equations with (N – 1) unknowns.
from the system of N equations. Then we extract a system of
(N – 2) equations with (N – 2) unknowns from the (N – 1)
equations. Continue this process until we get to and solve a
system of 2 equations.

11. Systems of Linear Equations With Three Variables
To solve for three unknowns, we need three pieces of
numerical information about the unknowns, i.e. three
sequations. The standard method for solving systems of
linear equations is the elimination method.
We use elimination method to extract a system of two
equations with two unknowns from the system of
three equations. Solve the system of 2 equations and plug
the answers back to get the third answer.
This is also the general method for solving a system of N
equations with N unknowns. We use elimination method to
extract a system of (N – 1) equations with (N – 1) unknowns.
from the system of N equations. Then we extract a system of
(N – 2) equations with (N – 2) unknowns from the (N – 1)
equations. Continue this process until we get to and solve a
system of 2 equations. Then plug the answers back to get
the other answers.

12. Systems of Linear Equations With Three Variables
Example A. We bought the following items:
2 hamburgers, 3 orders of fries and 3 sodas cost $13.
1 hamburger, 2 orders of fries and 2 sodas cost $8.
3 hamburgers, 2 fries, 3 sodas cost $13.
Find the price of each item.

13. Systems of Linear Equations With Three Variables
Example A. We bought the following items:
2 hamburgers, 3 orders of fries and 3 sodas cost $13.
1 hamburger, 2 orders of fries and 2 sodas cost $8.
3 hamburgers, 2 fries, 3 sodas cost $13.
Find the price of each item.
Let x = cost of a hamburger, y = cost of an order of fries,
z = cost of a soda.

14. Systems of Linear Equations With Three Variables
Example A. We bought the following items:
2 hamburgers, 3 orders of fries and 3 sodas cost $13.
1 hamburger, 2 orders of fries and 2 sodas cost $8.
3 hamburgers, 2 fries, 3 sodas cost $13.
Find the price of each item.
Let x = cost of a hamburger, y = cost of an order of fries,
z = cost of a soda.
2x + 3y + 3z = 13 E1

15. Systems of Linear Equations With Three Variables
Example A. We bought the following items:
2 hamburgers, 3 orders of fries and 3 sodas cost $13.
1 hamburger, 2 orders of fries and 2 sodas cost $8.
3 hamburgers, 2 fries, 3 sodas cost $13.
Find the price of each item.
Let x = cost of a hamburger, y = cost of an order of fries,
z = cost of a soda.
2x + 3y + 3z = 13 E1
x + 2y + 2z = 8 E2

16. Systems of Linear Equations With Three Variables
Example A. We bought the following items:
2 hamburgers, 3 orders of fries and 3 sodas cost $13.
1 hamburger, 2 orders of fries and 2 sodas cost $8.
3 hamburgers, 2 fries, 3 sodas cost $13.
Find the price of each item.
Let x = cost of a hamburger, y = cost of an order of fries,
z = cost of a soda.
{
2x + 3y + 3z = 13 E1
x + 2y + 2z = 8 E2
3x + 2y + 3z = 13 E3

17. Systems of Linear Equations With Three Variables
Example A. We bought the following items:
2 hamburgers, 3 orders of fries and 3 sodas cost $13.
1 hamburger, 2 orders of fries and 2 sodas cost $8.
3 hamburgers, 2 fries, 3 sodas cost $13.
Find the price of each item.
Let x = cost of a hamburger, y = cost of an order of fries,
z = cost of a soda.
{
2x + 3y + 3z = 13 E1
x + 2y + 2z = 8 E2
3x + 2y + 3z = 13 E3
Select x to eliminate since there is 1x in E2.

18. Systems of Linear Equations With Three Variables
Example A. We bought the following items:
2 hamburgers, 3 orders of fries and 3 sodas cost $13.
1 hamburger, 2 orders of fries and 2 sodas cost $8.
3 hamburgers, 2 fries, 3 sodas cost $13.
Find the price of each item.
Let x = cost of a hamburger, y = cost of an order of fries,
z = cost of a soda.
{
2x + 3y + 3z = 13 E1
x + 2y + 2z = 8 E2
3x + 2y + 3z = 13 E3
Select x to eliminate since there is 1x in E2.
–2*E 2 + E1:

19. Systems of Linear Equations With Three Variables
Example A. We bought the following items:
2 hamburgers, 3 orders of fries and 3 sodas cost $13.
1 hamburger, 2 orders of fries and 2 sodas cost $8.
3 hamburgers, 2 fries, 3 sodas cost $13.
Find the price of each item.
Let x = cost of a hamburger, y = cost of an order of fries,
z = cost of a soda.
{
2x + 3y + 3z = 13 E1
x + 2y + 2z = 8 E2
3x + 2y + 3z = 13 E3
Select x to eliminate since there is 1x in E2.
–2*E 2 + E1:
–2x – 4y – 4z = -16

20. Systems of Linear Equations With Three Variables
Example A. We bought the following items:
2 hamburgers, 3 orders of fries and 3 sodas cost $13.
1 hamburger, 2 orders of fries and 2 sodas cost $8.
3 hamburgers, 2 fries, 3 sodas cost $13.
Find the price of each item.
Let x = cost of a hamburger, y = cost of an order of fries,
z = cost of a soda.
{
2x + 3y + 3z = 13 E1
x + 2y + 2z = 8 E2
3x + 2y + 3z = 13 E3
Select x to eliminate since there is 1x in E2.
–2*E 2 + E1:
–2x – 4y – 4z = -16
+) 2x + 3y + 3z = 13

21. Systems of Linear Equations With Three Variables
Example A. We bought the following items:
2 hamburgers, 3 orders of fries and 3 sodas cost $13.
1 hamburger, 2 orders of fries and 2 sodas cost $8.
3 hamburgers, 2 fries, 3 sodas cost $13.
Find the price of each item.
Let x = cost of a hamburger, y = cost of an order of fries,
z = cost of a soda.
{
2x + 3y + 3z = 13 E1
x + 2y + 2z = 8 E2
3x + 2y + 3z = 13 E3
Select x to eliminate since there is 1x in E2.
–2*E 2 + E1:
–2x – 4y – 4z = -16
+) 2x + 3y + 3z = 13
0– y – z =–3

22. Systems of Linear Equations With Three Variables
Example A. We bought the following items:
2 hamburgers, 3 orders of fries and 3 sodas cost $13.
1 hamburger, 2 orders of fries and 2 sodas cost $8.
3 hamburgers, 2 fries, 3 sodas cost $13.
Find the price of each item.
Let x = cost of a hamburger, y = cost of an order of fries,
z = cost of a soda.
{
2x + 3y + 3z = 13 E1
x + 2y + 2z = 8 E2
3x + 2y + 3z = 13 E3
Select x to eliminate since there is 1x in E2.
–2*E 2 + E1: –3*E 2 + E3:
–2x – 4y – 4z = -16
+) 2x + 3y + 3z = 13
0– y – z =–3

23. Systems of Linear Equations With Three Variables
Example A. We bought the following items:
2 hamburgers, 3 orders of fries and 3 sodas cost $13.
1 hamburger, 2 orders of fries and 2 sodas cost $8.
3 hamburgers, 2 fries, 3 sodas cost $13.
Find the price of each item.
Let x = cost of a hamburger, y = cost of an order of fries,
z = cost of a soda.
{
2x + 3y + 3z = 13 E1
x + 2y + 2z = 8 E2
3x + 2y + 3z = 13 E3
Select x to eliminate since there is 1x in E2.
–2*E 2 + E1: –3*E 2 + E3:
–2x – 4y – 4z = -16 –3x – 6y – 6z = –24
+) 2x + 3y + 3z = 13
0– y – z =–3

24. Systems of Linear Equations With Three Variables
Example A. We bought the following items:
2 hamburgers, 3 orders of fries and 3 sodas cost $13.
1 hamburger, 2 orders of fries and 2 sodas cost $8.
3 hamburgers, 2 fries, 3 sodas cost $13.
Find the price of each item.
Let x = cost of a hamburger, y = cost of an order of fries,
z = cost of a soda.
{
2x + 3y + 3z = 13 E1
x + 2y + 2z = 8 E2
3x + 2y + 3z = 13 E3
Select x to eliminate since there is 1x in E2.
–2*E 2 + E1: –3*E 2 + E3:
–2x – 4y – 4z = -16 –3x – 6y – 6z = –24
+) 2x + 3y + 3z = 13 +) 3x + 2y + 3z = 13
0– y – z =–3

25. Systems of Linear Equations With Three Variables
Example A. We bought the following items:
2 hamburgers, 3 orders of fries and 3 sodas cost $13.
1 hamburger, 2 orders of fries and 2 sodas cost $8.
3 hamburgers, 2 fries, 3 sodas cost $13.
Find the price of each item.
Let x = cost of a hamburger, y = cost of an order of fries,
z = cost of a soda.
{
2x + 3y + 3z = 13 E1
x + 2y + 2z = 8 E2
3x + 2y + 3z = 13 E3
Select x to eliminate since there is 1x in E2.
–2*E 2 + E1: –3*E 2 + E3:
–2x – 4y – 4z = -16 –3x – 6y – 6z = –24
+) 2x + 3y + 3z = 13 +) 3x + 2y + 3z = 13
0– y – z =–3 0 – 4y – 3z = –11

26. Systems of Linear Equations With Three Variables
Example A. We bought the following items:
2 hamburgers, 3 orders of fries and 3 sodas cost $13.
1 hamburger, 2 orders of fries and 2 sodas cost $8.
3 hamburgers, 2 fries, 3 sodas cost $13.
Find the price of each item.
Let x = cost of a hamburger, y = cost of an order of fries,
z = cost of a soda.
{
2x + 3y + 3z = 13 E1
x + 2y + 2z = 8 E2
3x + 2y + 3z = 13 E3
Select x to eliminate since there is 1x in E2.
–2*E 2 + E1: –3*E 2 + E3:
–2x – 4y – 4z = -16 –3x – 6y – 6z = –24
+) 2x + 3y + 3z = 13 +) 3x + 2y + 3z = 13
0– y – z =–3 0 – 4y – 3z = –11
Group these two equations into a system.

27. Systems of Linear Equations With Three Variables
Hence, we've reduced the original system to two equations
with two unknowns:
–y–z =–3
{–4y – 3z = –11

28. Systems of Linear Equations With Three Variables
Hence, we've reduced the original system to two equations
with two unknowns:
–y–z =–3 (-1)
{–4y – 3z = –11

29. Systems of Linear Equations With Three Variables
Hence, we've reduced the original system to two equations
with two unknowns:
E4
{ –y–z =–3
–4y – 3z = –11
(-1)
{ y +z = 3
4y + 3z = 11 E5

30. Systems of Linear Equations With Three Variables
Hence, we've reduced the original system to two equations
with two unknowns:
E4
{ –y–z =–3
–4y – 3z = –11
(-1)
{ y +z = 3
4y + 3z = 11 E5
To eliminate z we –3*E 4 + E5:

31. Systems of Linear Equations With Three Variables
Hence, we've reduced the original system to two equations
with two unknowns:
E4
{ –y–z =–3
–4y – 3z = –11
(-1)
{ y +z = 3
4y + 3z = 11 E5
To eliminate z we –3*E 4 + E5:
–3y – 3z = –9

32. Systems of Linear Equations With Three Variables
Hence, we've reduced the original system to two equations
with two unknowns:
E4
{ –y–z =–3
–4y – 3z = –11
(-1)
{ y +z = 3
4y + 3z = 11 E5
To eliminate z we –3*E 4 + E5:
–3y – 3z = –9
+) 4y + 3z = 11

33. Systems of Linear Equations With Three Variables
Hence, we've reduced the original system to two equations
with two unknowns:
E4
{ –y–z =–3
–4y – 3z = –11
(-1)
{ y +z = 3
4y + 3z = 11 E5
To eliminate z we –3*E 4 + E5:
–3y – 3z = –9
+) 4y + 3z = 11
y+0 = 2
y= 2

34. Systems of Linear Equations With Three Variables
Hence, we've reduced the original system to two equations
with two unknowns:
E4
{ –y–z =–3
–4y – 3z = –11
(-1)
{ y +z = 3
4y + 3z = 11 E5
To eliminate z we –3*E 4 + E5:
–3y – 3z = –9
+) 4y + 3z = 11
y+0 = 2
y= 2
To get z, set 2 for y in E4:

35. Systems of Linear Equations With Three Variables
Hence, we've reduced the original system to two equations
with two unknowns:
E4
{ –y–z =–3
–4y – 3z = –11
(-1)
{ y +z = 3
4y + 3z = 11 E5
To eliminate z we –3*E 4 + E5:
–3y – 3z = –9
+) 4y + 3z = 11
y+0 = 2
y= 2
To get z, set 2 for y in E4:
2+z=3

36. Systems of Linear Equations With Three Variables
Hence, we've reduced the original system to two equations
with two unknowns:
E4
{ –y–z =–3
–4y – 3z = –11
(-1)
{ y +z = 3
4y + 3z = 11 E5
To eliminate z we –3*E 4 + E5:
–3y – 3z = –9
+) 4y + 3z = 11
y+0 = 2
y= 2
To get z, set 2 for y in E4:
2+z=3 z=1

37. Systems of Linear Equations With Three Variables
Hence, we've reduced the original system to two equations
with two unknowns:
E4
{ –y–z =–3
–4y – 3z = –11
(-1)
{ y +z = 3
4y + 3z = 11 E5
To eliminate z we –3*E 4 + E5:
–3y – 3z = –9
+) 4y + 3z = 11
y+0 = 2
y= 2
To get z, set 2 for y in E4:
2+z=3 z=1
For x, set 2 for y , set 1 for z in E2: x + 2y + 2z = 8

38. Systems of Linear Equations With Three Variables
Hence, we've reduced the original system to two equations
with two unknowns:
E4
{ –y–z =–3
–4y – 3z = –11
(-1)
{ y +z = 3
4y + 3z = 11 E5
To eliminate z we –3*E 4 + E5:
–3y – 3z = –9
+) 4y + 3z = 11
y+0 = 2
y= 2
To get z, set 2 for y in E4:
2+z=3 z=1
For x, set 2 for y , set 1 for z in E2: x + 2y + 2z = 8
x + 2(2) + 2(1) = 8

39. Systems of Linear Equations With Three Variables
Hence, we've reduced the original system to two equations
with two unknowns:
E4
{ –y–z =–3
–4y – 3z = –11
(-1)
{ y +z = 3
4y + 3z = 11 E5
To eliminate z we –3*E 4 + E5:
–3y – 3z = –9
+) 4y + 3z = 11
y+0 = 2
y= 2
To get z, set 2 for y in E4:
2+z=3 z=1
For x, set 2 for y , set 1 for z in E2: x + 2y + 2z = 8
x + 2(2) + 2(1) = 8
x+6 =8

40. Systems of Linear Equations With Three Variables
Hence, we've reduced the original system to two equations
with two unknowns:
E4
{ –y–z =–3
–4y – 3z = –11
(-1)
{ y +z = 3
4y + 3z = 11 E5
To eliminate z we –3*E 4 + E5:
–3y – 3z = –9
+) 4y + 3z = 11
y+0 = 2
y= 2
To get z, set 2 for y in E4:
2+z=3 z=1
For x, set 2 for y , set 1 for z in E2: x + 2y + 2z = 8
x + 2(2) + 2(1) = 8
x+6 =8
x=2

41. Systems of Linear Equations With Three Variables
Hence, we've reduced the original system to two equations
with two unknowns:
E4
{ –y–z =–3
–4y – 3z = –11
(-1)
{ y +z = 3
4y + 3z = 11 E5
To eliminate z we –3*E 4 + E5:
–3y – 3z = –9
+) 4y + 3z = 11
y+0 = 2
y= 2
To get z, set 2 for y in E4:
2+z=3 z=1
For x, set 2 for y , set 1 for z in E2: x + 2y + 2z = 8
x + 2(2) + 2(1) = 8
x+6 =8
x=2
Hence the solution is (2, 2, 1).