Warm Up
• Write down objective and homework in agenda
• Lay out homework (none)• Homework (Systems of Equations graphing)
• WELCOME BACK! • Get a calculator!
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
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.
2Slope = = 2
1y-intercept = 2
y-intercept = -1
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.
2Slope = = 2
1y-intercept = -1
Solving a system of equations by graphing.
There are 3 steps to solving a system using a graph.
Step 1: Graph both equations.
Step 2: Do the graphs intersect?
Step 3: Check your solution.
Graph using slope and y – intercept or x- and y-intercepts. Be sure to use a ruler and graph paper!
This is the solution! LABEL the solution!
Substitute the x and y values into both equations to verify the point is a solution to both equations.
What is the solution of the system graphed below?
1. (2, -2)2. (-2, 2)3. No solution4. Infinitely many solutions
Graph the equations.
2x + y = 4(0, 4) and (2, 0)
x - y = 2(0, -2) and (2, 0)
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 – 3m = 2 and b = -3
y = 2x + 1m = 2 and b = 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 4• 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
• 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)
Extra Resources
• http://www.phschool.com/webcodes10/index.cfm?fuseaction=home.gotoWebCode&wcprefix=ata&wcsuffix=0701
• http://www.regentsprep.org/Regents/math/ALGEBRA/AE3/PracGr.htm
• http://www.regentsprep.org/Regents/math/ALGEBRA/AE3/GrSys.htm
• http://www.quia.com/cz/43456.html?AP_rand=1117418363
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 calculatorExample
• 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)
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