Warm Up

• Write down objective and homework in agenda

• Lay out homework (none)• Homework (Systems of Equations graphing)

• WELCOME BACK! • Get a calculator!

Warm UpGraph 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

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)

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