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

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 y = -3 – 3x
(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)
(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 (watch this video to see it done by a person):
“Y=” to put in your equations
Press “GRAPH” (ZoomOut if necessary)
CALC menu (2nd “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: ___________
+20
+20
+$5
+$15
Company T
5/20 = $0.25 per minute
15/20 = $0.75 per minute

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