1. Solving Systems Using Elimination
Section 6-3 Part 2

2. Further Elimination
In Part 1 of this lesson, you was that to eliminate a variable, its coefficients must have a sum or difference of zero.
In some cases, you will first need to multiply one or both of the equations by a number so that one variable has opposite coefficients, so you can add or subtract to eliminate the variable.

3. Example: Multiplying One Equation
2x + 2y = 6
3x – y = 5

4. Example: Continued
2x + 2y = 6
3x – y = 5

5. Example: Continued
2x + 2y = 6
3x – y = 5

6. Example: Multiplying One Equation
x + 4y = 7
4x – 3y = 9

7. Example: Continued
x + 4y = 7
4x – 3y = 9

8. Your Turn:
Solve the system by elimination.
x + 2y = 11
–3x + y = –5

9. Your Turn:
Solve the system by elimination.
3x + 2y = 6
–x + y = –2

10. Example: Multiplying Both Equations
3x + 4y = -1
4x – 3y = 7

11. Your Turn:
Solve the system by elimination.
–5x + 2y = 32
2x + 3y = 10

12. Your Turn: Solve the system by elimination. 2x + 5y = 26

13. infinitely many solutions
IDENTIFYING THE NUMBER OF SOLUTIONS
NUMBER OF SOLUTIONS OF A LINEAR SYSTEM
CONCEPT
SUMMARY y x y x y x
Lines intersect
one solution
Lines are parallel
no solution
Lines coincide
infinitely many solutions

14. Identifying The Number of Solutions
If both variable terms are eliminated as you solve a system of equations, the answer is either no solution or infinite solutions.
No solution: get a false statement when solving the system.
Infinite solutions: get a true statement when solving the system.

15. A Linear System with Infinite Solutions
Show that this linear system
has infinitely many solutions.
– 2 x  y  3 Equation 1
– 4 x  2y  6 Equation 2
METHOD: Elimination

16. Show that this linear system
A Linear System with No Solution
Show that this linear system
has no solution.
2 x  y  Equation 1
2 x  y  Equation 2
METHOD: Elimination

17. Your Turn:
Solve the systems using elimination.

18. Summary

19. Summary of Methods for Solving Systems
7.3 The Elimination Method
Summary of Methods for Solving Systems
Example
Suggested Method
Why
6x + y = 10
y = 5
Substitution
The value of one variable is known and can easily be substituted into the other equation.

20. Summary of Methods for Solving Systems
7.3 The Elimination Method
Summary of Methods for Solving Systems
Example
Suggested Method
Why
2x – 5y = –20
4x + 5y = 14
Elimination
eliminate ‘y’  5
Add the two equations

21. Summary of Methods for Solving Systems
7.3 The Elimination Method
Summary of Methods for Solving Systems
Example
Suggested Method
Why
9a – 2b = –11
8a + 4b = 25
Elimination
eliminate ‘b’  4
Multiply first equation by 2
Add the equations