1. Solving Systems by Graphing or Substitution.
Objective: To solve a system of linear equations in two variables by graphing or by substitution.

2. System of Equations
A system of equations is a collection of equations in the same variables.

3. System of Equations
A system of equations is a collection of equations in the same variables.
The solution of a system of two linear equations in x and y is any ordered pair, (x, y), that satisfies both equations. The solution (x, y) is also the point of intersection for the graphs of the lines in the system. For example, the ordered pair (2, -1) is the solution of the system below.

7. Example 1

8. Example 1

9. Example 1

10. Try This
Graph and classify the following system. Then, find the solution from the graph.

11. Try This
Graph and classify the following system. Then, find the solution from the graph.
Consistent, Independent.

12. Try This
Graph and classify the following system. Then, find the solution from the graph.
Consistent, Independent.
Solution is the point (0, 4)

13. Substitution
There is also another way to find solutions to a system of equations. This is called substitution.

14. Example 2

15. Example 2

16. Example 2

17. Example 2

18. Try This
Use substitution to solve the system. Check your solution.

19. Try This Use substitution to solve the system. Check your solution.
Solve the first equation for y and substitute it into the second equation.

20. Try This Use substitution to solve the system. Check your solution.
Solve the first equation for y and substitute it into the second equation. Find y. Use either equation.

21. Example 3
A laboratory technician is mixing a 10% saline solution with a 4% saline solution. How much of each solution is needed to make 500 milliliters of a 6% solution?

22. Example 3
A laboratory technician is mixing a 10% saline solution with a 4% saline solution. How much of each solution is needed to make 500 milliliters of a 6% solution?

23. Example 3

24. Example 3

25. Example 3

26. Example 3

27. Try This
If a 7% saline solution and a 4% saline solution are mixed to make 500 milliliters of a 5% solution, how much of each solution, to the nearest milliliter, is needed?

28. Try This
If a 7% saline solution and a 4% saline solution are mixed to make 500 milliliters of a 5% solution, how much of each solution, to the nearest milliliter, is needed?
You need two equations.
x represents the 7% solution.
y represents the 4% solution.

29. Try This
If a 7% saline solution and a 4% saline solution are mixed to make 500 milliliters of a 5% solution, how much of each solution, to the nearest milliliter, is needed?
You need two equations.
x represents the 7% solution.
y represents the 4% solution.
Solve the first equation for y
and substitute.

30. Try This
If a 7% saline solution and a 4% saline solution are mixed to make 500 milliliters of a 5% solution, how much of each solution, to the nearest milliliter, is needed?
Solve.

31. Example 4

32. Example 4

33. Example 4

34. Example 4

35. Example 4

37. Homework
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You need to do all of these problems to be good at this. Please do that.