Warm Up Write down objective and homework in agenda Lay out homework (none) Homework (Systems of Equations graphing) WELCOME BACK!  Get a calculator!

Warm Up Graph the following lines

Vocabulary system of linear equations: two or more linear equations graphed in the same coordinate plane solution of a system of linear equations: any ordered pair in a system that makes all the equations true no solution: when two lines are parallel, there are no points of intersection infinitely many solutions: when the graphs of the systems of equations are the same line

What is a system of equations? A system of equations is when you have two or more equations using the same variables. The solution to the system is the point that satisfies ALL of the equations. This point will be an ordered pair. When graphing, you will encounter three possibilities.

A SYSTEM of equations A system can have one solution, infinitely many solutions, or no solutions. One solution means the lines intersect No solutions means the lines never touch (parallel) Many solutions means the lines are the same Draw diagram!

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

Parallel Lines These lines never intersect! Since the lines never cross, there is NO SOLUTION! Parallel lines have the same slope with different y-intercepts.

Coinciding Lines These lines are the same! Since the lines are on top of each other, there are INFINITELY MANY SOLUTIONS! Coinciding lines have the same slope and y-intercepts.

Solving a system of equations by graphing. There are 3 steps to solving a system using a graph. Graph using slope and y – intercept or x- and y-intercepts. Be sure to use a ruler and graph paper! Step 1: Graph both equations. This is the solution! LABEL the solution! Step 2: Do the graphs intersect? Substitute the x and y values into both equations to verify the point is a solution to both equations. Step 3: Check your solution.

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

Graph the equations. 2x + y = 4 (0, 4) and (2, 0) x - y = 2 Where do the lines intersect? (2, 0) 2x + y = 4 x – y = 2

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

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!

Example Graph the following equations on the same graph and find the solution to the system of equations: y = 2x -7 y = 1 a) What solution did you find? (4, 1) b) To check: Plug your solution(s) into each equation to see if it works. c) Consider the point (5, 3). Does it work in the 1st equation? The 2nd? Is this a solution to the system? First but not second, so not a system!

Example Graph and find the solution: y = 3 x - 6 4 y = 3x + 1 What is the solution to this system? NO solution  Explain why you came to this conclusion.

Example Graph and find the solution: 2x + y = 6 y = -2x + 6 What is the solution to this system? Coinciding lines! All solutions  Explain why you came to this conclusion.

You Try! Graph the following equations on the same graph and find the solution to the system of equations: y = -2x + 3 y = 1/2x - 2 a) What solution did you find? (2, -1) b) To check: Plug your solution(s) into each equation to see if it works.

You Try! Graph the following equations on the same graph and find the solution to the system of equations: y = 3x + 5 y = 3x - 2 a) What solution did you find? NO Solution! b) To check: Plug your solution(s) into each equation to see if it works.

Graph and find the solution: 2x + y = 6 y = -2x + 6 What is the solution to this system? Infinite Solutions  Explain why you came to this conclusion. They are coinciding lines, which means they intersect at every point; therefore there is infinite solutions

a) What solution did you find? (9,11) 1. Graph the following equations on the same graph and find the solution to the system of equations: y = 2x – 7 y = x + 2 a) What solution did you find?   (9,11) b ) To check: Plug your solution(s) into each equation to see if it works.   c) Consider the point (5, 3). Does it work in the 1st equation? the 2nd? Is this a solution to the system? It works in the 1st, but not 2nd; therefore it’s NOT a solution.

Graph and find the solution: y = x – 6 y = x + 1   Graph and find the solution: y = x – 6  y = x + 1   What is the solution to this system? No solution! Explain why you came to this conclusion. They are parallel and will never intersect

What is the solution of this system? 3x – y = 8 2y = 6x -16 (3, 1) (4, 4) No solution Infinitely many solutions

Summarize the number of possible solutions to a system of two equations in two variables and explain how each possibility could occur. (There are THREE)

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Steps to Graphing Systems on Calc. Step 1: Press y =, clear out old equations and enter new Step 2: Press GRAPH Step 3:Find the intersection of the 2 equations by pressing 2nd CALC (over TRACE) 5: intersect. Press ENTER 3 times to find the intersection

Solving systems on graphing calculator Example ALWAYS put in slope intercept form FIRST Step 1: Press y = and clear any old equations. Enter: y = x + 6 y = 2x + 4

Step 2 Step 2: Press GRAPH (If the intersection is off the graph, press ZOOM; arrow down until you see 0: ZoomFit and hit enter or you can adjust the window.) If you can’t see the intersection on your screen, the calculator won’t find it!!!

Step 3: Find the intersection of the 2 equations by pressing 2nd CALC (over TRACE) 5: intersect. Press ENTER 3 times to find the intersection The intersection is (2, 8). *Check the point of intersection by substituting the x- and y-values into both equations

  Examples Find the intersection of each system of equations by using a graphing calculator. Check your solutions. (Hint: Sometimes you have to solve for y first.) 1. y = −4x −1 2. y =3x −4 y = − x + 2 y=5x−12 Answers: 1. (-1, 3) 2. (4, 8)

Practice No Solution (5, 5) 5

Practice 3. 2x−4y=8 4. x + y = −6 x − y = 4 x −5y =0 3. (4, 0) 4. (-5, -1)

You Try! Hint: Sometimes you have to solve for y first.

Answers 1. (-1, 3) 2. (4, 8) 3. (4, 0) 4. (-5, -1)

Extra Resources http://teachers.henrico.k12.va.us/math/hcpsalgebra1/Documents/examviewweb/ev9-1.htm http://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-228s.html