6.1 Solving Systems Of Equations By Graphing

Bell Work Evaluate each expression for x = 1 and y = –3. 1. x – 4y 2. –2x + y Write each expression in slope-intercept form. 3. y – x = 1 4. 2x + 3y = 6 5.

Cheetah Vs. Usain Bolt Video: http://youtu.be/U6K-F9Bjl8s

Distance (5.95, 100) Bolt= Cheetah= 100 m (9.58, 100) 9 10 Time

Distance Bolt= Cheetah= 100 m (9.58, 100) 40 m 9 10 Time

Solving Linear Equations By Graphing Our goal is to; Graph TWO lines on same graph Locate where they intersect (or if they do at all)

Solving a Linear Equation Linear equations can look like; y = 3x + 7 Or -3x + y = 7 7 + 3x = y …

3 Graphing Possibilities Intersecting Lines Parallel Lines Coinciding Lines

1) Intersecting Lines The point where the lines intersect is your solution. The solution of this graph is (1, 2) (1,2)

What is the solution of the system graphed below? (2, -2) (-2, 2) No solution Infinitely many solutions

Where do these graphs intersect

2) Parallel Lines These lines never intersect! Since the lines never cross, there is NO SOLUTION! Parallel lines = SAME SLOPE, different y-int.

Our Slopes Must Be Twins!!! Parallel Lines Hi, I’m Two Hey, I’m Two Too! Our Slopes Must Be Twins!!!

What is the solution of this system? y = 3x + 3 y = 3x - 12 (3, -12) Parallel solutions No solution Infinitely many solutions

3) Coinciding Lines These lines are the same! Since the lines are on top of each other, there are INFINITELY MANY SOLUTIONS! Coinciding lines = SAME SLOPE, SAME Y-INT.

What is the solution of this system? y = 3x - 8 (3, 1) (4, 4) No solution Infinitely many solutions

Problem-Solving Application Sally and Jenni are reading the same book. Wren is on page 14 and reads 2 pages every night. Jenni is on page 6 and reads 3 pages every night. After how many nights will they have read the same number of pages? How many pages will that be? pages Time

 (8, 30) Nights

Lets try one: Find the solution to the following system: y = -2x+4 y = x -2 m=-2 b=4 m=1 b=-2 Step 1:Graph both equations Step 2: Find Where the lines intersect? THE SOLUTION: (2,0)

Check your answer! Y=-2x+4 2(2) + 4 = 0 y = x-2 (2) – 2 = 0 To check your answer, substitute the point back into both equations. Y=-2x+4 2(2) + 4 = 0 y = x-2 (2) – 2 = 0

You try: Where do the lines intersect? y = 2x – 1 y = –x + 5 Graph both equations Where do they intersect? The point is (2, 3)

Try again: Find the solution to the system y = -2x-1 y = -2x+4 Graph both equations Where do they intersect? NO SOLUTION

Find the Solution

Summary Intersecting Lines have how many solutions? Parallel Lines have the same what? How can you tell if two equations are coinciding without graphing them? Do you have to graph the system to know the answer if both equations have the same slope?

HOMEWORK Find an example in real life (just like Usain Bolt vs. Cheetah) and graph it. I want to see the following What two things are you graphing What is your x-axis and y-axis Graph them Do they intersect? Where?

Challenge: Find the solution to the following system: y = 2x – 3 -2x + y = 1 Graph both equations. Put both equations in slope-intercept y=mx+b y = 2x + 1 Graph using slope and y-intercept

Graph the equations. y = 2x – 3 m = 2 and b = -3 y = 2x + 1 Where do the lines intersect? No solution! Notice that the slopes are the same with different y-intercepts. If you recognize this early, you don’t have to graph them!