1. Introduction to Systems of Equations (and Solving by Graphing) Unit 5 Day 3

2. Systems Two or more linear equations together form a system of linear equations. Example: A solution of the system of linear equations is any ordered pair that makes all the equations true. There are several ways to find the solution for a system of linear equations. Today, we will explore how to solve by graphing.

3. Example One Looking at these graphs of a system of equations, where do you think the solution is? Solution: _______ Solution: _______Solution: _______ (1, 3) (-1, -2)none

4. There are 3 types of solutions ONE SOLUTION When the lines intersects at one point. (x, y)

5. There are 3 types of solutions NO SOLUTION When the lines are parallel and never intersect

6. There are 3 types of solutions INFINITELY MANY SOLUTIONS When the two equations graph the same exact line.

7. Example Two a.) y = -1/2x + 2 3x + y = -3 (subtract 3x to solve for y) y = -3 – 3x Solution: ( -2, 3 )

8. Example Two: b.) y = 2x + 1 -12x + 6y = 6 (add 12x to both sides) 6y = 6 + 12x (divide by 6 to both sides) y = 1 + 2x SAME LINE! Infinitely many solutions

9. Example Two: c.) y = x + 5 y = -4x Let’s check your answer by calculator: – “Y=” to put in your equations – Press “GRAPH” (ZoomOut if necessary) – CALC menu (2 nd “TRACE”) – Option 5: intersect – ENTER : 3 times SOLUTION: (-1, 4)

10. Example Three This graph shows the money charged given number of minutes used. How much does Company S charge as a starting fee? _____ How much does Company T charge as a starting fee? _____ $15 $5

11. Example Three This graph shows the money charged given number of minutes used. For how many minutes do the companies charge the same amount? ____ What is the charge for this? ____ 20 minutes $20

12. Example Three This graph shows the money charged given number of minutes used. Which company charges more per minute? _________ Company S price per minute: ___________ Company T price per minute: ___________ Company T +$5 +20 5/20 = $0.25 per minute 15/20 = $0.75 per minute +$15 +20

13. Example Three This graph shows the money charged given number of minutes used. Write an equation for Company S:______________ Company T: ______________ When would it be better to use Company T instead of Company S? y = $0.25x + 15 y = $0.75x + 5 When you use less than 20 minutes - because over 20 minutes, T is more expensive