Math 71A

1. Math 71A 3.1 – Systems of Linear Equations in Two Variables

2. Suppose we have two linear equations: Together, they make a ________________________________ (also called a ___________________). Solutions must satisfy all equations. For example, is a solution to the above system since it satisfies both equations.

3. Suppose we have two linear equations: Together, they make a ________________________________ (also called a ___________________). Solutions must satisfy all equations. For example, is a solution to the above system since it satisfies both equations. system of linear equations

4. Suppose we have two linear equations: Together, they make a ________________________________ (also called a ___________________). Solutions must satisfy all equations. For example, is a solution to the above system since it satisfies both equations. system of linear equations linear system

5. Suppose we have two linear equations: Together, they make a ________________________________ (also called a ___________________). Solutions must satisfy all equations. For example, is a solution to the above system since it satisfies both equations. system of linear equations linear system

6. Suppose we have two linear equations: Together, they make a ________________________________ (also called a ___________________). Solutions must satisfy all equations. For example, is a solution to the above system since it satisfies both equations. system of linear equations linear system

7. Ex 1.Determine if is a solution to the following system: Is a solution?

8. Graphing Systems of Linear Equations Besides , are there any other solutions to the following system? We can show this by graphing the two lines.

9. Graphing Systems of Linear Equations

10. Graphing Systems of Linear Equations

11. Graphing Systems of Linear Equations

12. Graphing Systems of Linear Equations

13. Graphing Systems of Linear Equations

14. Graphing Systems of Linear Equations

15. Graphing Systems of Linear Equations

16. Graphing Systems of Linear Equations is the only solution to system

17. Substitution Method Ex 2.Solve by the substitution method:

18. Substitution Method Solve one equation for or Substitute the expression we get for or into the other equation This makes an equation with one variable that we can solve After that, we can find the other variable

19. Substitution Method Solve one equation for or Substitute the expression we get for or into the other equation This makes an equation with one variable that we can solve After that, we can find the other variable

20. Substitution Method Solve one equation for or Substitute the expression we get for or into the other equation This makes an equation with one variable that we can solve After that, we can find the other variable

21. Substitution Method Solve one equation for or Substitute the expression we get for or into the other equation This makes an equation with one variable that we can solve After that, we can find the other variable

22. Substitution Method Ex 3.Solve by the substitution method:

23. Addition Method Ex 4.Solve by the addition method:

24. Addition Method Ex 5.Solve by the addition method (Hint: clear the fractions first!):

25. No Solution or Infinitely Many Solutions No solution Exactly one solution Infinitely many solutions

26. No Solution or Infinitely Many Solutions No solution Exactly one solution Infinitely many solutions

27. No Solution or Infinitely Many Solutions No solution Exactly one solution Infinitely many solutions

28. No Solution or Infinitely Many Solutions No solution Exactly one solution Infinitely many solutions

29. No Solution or Infinitely Many Solutions No solution Exactly one solution Infinitely many solutions

30. No Solution or Infinitely Many Solutions intersect once identical parallel No solution Exactly one solution Infinitely many solutions

31. No Solution or Infinitely Many Solutions Ex 6.Solve the system:

32. No Solution or Infinitely Many Solutions Ex 7.Solve the system: