1. 3.1 - Solving Systems by Graphing

2. All I do is Solve!

3.  Graph the following pairs of equations:
y = x + 5
y = -2x + 5
b. y = 3x + 2
y = 3x - 1
c. y = -4x - 2
y = 8x + 4 -2

4. System of Equations: A set of two or more equations that use the same variables. Solution to a System: A set of values that makes ALL equations true.

5. Is (-3, 4) a solution to the system?

6. Is (3,1) a solution for the following system?

7. Types of Solutions
Intersecting Lines have ONE unique solution.
Coincidental Lines (or same lines) have MANY solutions.
Parallel Lines have NO solutions!

8. Solve by Graphing:

9. Graphing Calculator

10. Solve the following systems by graphing:

11. Solving by Graphing:

12. Classwork (To Be Turned In):
What type of solution does each system have? If the solution exist, what is it? 2) 1) 3) 4)

13. 3.2 Solving Systems Algebraically

14. Solving Systems by Substitution

15. The Substitution Method
Not every system can be solved easily by graphing. Sometimes it is not always clear from the graph where the solution is.
We can use an algebraic method called SUBSTITUTION to find the exact solution without a graphing calculator.

16. Solving by Substitution
Solve for one of the variables.
Substitute the expression of the equation you solved for into the other equation.
Solve for the variable.
Substitute the value of x into either equation and solve.

17. Solve the system by substitution.

18. You Try! Solve the system by substitution

19. Solving Systems by Elimination

20. Elimination
We can solve by elimination by either Adding or Subtracting two equations to eliminate a variable!

21. Solve by Elimination:

22. Solve by Elimination:

23. You Try! Solve the systems by elimination:

24. Note: Sometimes with elimination you will have to multiply one or both of the equations in a system. This creates an EQUIVALENT SYSTEM that has the same solution to the original.

25. Solve the system by elimination.

26. Special Solutions
Solve each system by elimination

27. 3.6 Solving Systems with Three Variables

28. 3-variable Systems
Systems with 3 variables will have 3 equations. These type of systems are in three dimensions! So it is not going to be easy to find their solution by graphing.

29. We can solve Systems with 3 variables, using Elimination OR Substitution.

30. Solve by Elimination

31. SO MUCH WORK!!!
Luckily, we have an easier way to do this! When solving system of the equations we can use Matrices!!

32. Writing Systems as a Matrix Equation
For Matrix Equations in the form
AX = B
A is called the COEFFICIENT MATRIX
X is called the VARIABLE MATRIX
B is called the CONSTANT MATRIX

33. A Coefficient is a number INFRONT of a variable.
A Variable is a value represented by a letter or symbol
A Constant is a number WITHOUT a variable.

34. Write the following System as a Matrix:

35. Remember that when solving matrix equations:
If AX=B then X = A-1B

36. Solve the Matrix Equation
(4,-10,1)

37. Write as a Matrix Equations and Solve!
(1.25, 2.5, -1.75)

38. Write the follow systems as Matrix Equations. Then Solve! 1. 2.
(-1,0,3) (4,3,4)

39. Unique Solutions
Remember, Systems can have 1 solution, NO solutions, or MANY solutions. IF matrix A’s Determinate is 0 then the matrix does NOT have an inverse and the systems does NOT have a unique solution. IF matrix A’s Determine is NOT 0 then the matrix has an inverse and the system has a unique solutions!

40. Unique Solution
Determine if there is a unique solution.

41. Example
The sum of three numbers is 12. The 1st is 5 times the 2nd. The sum of the 1st and 3rd is 9. Find the numbers.
Represent as a system
Represent as a matrix -> majority of the time, easiest to solve as a system
(15, 3, -6)

42. HW 3.6/ Classwork
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