Get Out YourChapter 5 Notes

• 5.1• 5.2• 5.3• 5.4• 5.5

LearningLearningObjective:Objective:

Defining “System” of

Equations• A grouping of 2 or more equations, containing one or more variables.

x + y = 2x + y = 2

2x + y = 52x + y = 5

How do we “solve” a system of equations?

• By finding the point where two or more equations intersect

x + y = 6x + y = 6

y = 2xy = 2x Point of intersectionPoint of intersection

66

44

22

11 22

(2,4)(2,4)

There are 3 methods that we can use to Solve

a “System” of Equations

Today we are going to Today we are going to focus on solving by focus on solving by graphing!graphing!

TODAY WE ARE GOING TO FOCUS ON “GRAPHING”

TO SOLVE A SYSTEM OF

EQUATIONS

System of Equations:“One Solution”

One SolutionOne Solution: : • the lines of two equations intersectthe lines of two equations intersect

The The solutionsolution is the is the POINTPOINT that they that they intersect at.intersect at.

1)1)Decide what form the equations are Decide what form the equations are in:in:• What are the three?What are the three?

A. A. B.B.C. C.

2) Graph both lines.2) Graph both lines.

3) Determine the 3) Determine the point of point of intersectionintersection and write this point as and write this point as an ordered pair.an ordered pair.

Slope-Intercept FormSlope-Intercept Form

Standard FormStandard Form

Point-Slope FormPoint-Slope Form

Graph the system of equations. Determine whether Graph the system of equations. Determine whether the system has one solution, no solution, or the system has one solution, no solution, or infinitely many solutions. If the system has one infinitely many solutions. If the system has one solution, determine the solution.solution, determine the solution.

y= x - 2y= x - 2 y x 2

33

Step 1- what form are the linear equations in?Step 1- what form are the linear equations in?

Slope-Intercept FormSlope-Intercept Form

Use the slope and the y-Use the slope and the y-intercept to graph.intercept to graph.

y = x – 2y = x – 2 y x 2

33

Step 2Step 2: Use the slope and y-intercept of each line to plot two : Use the slope and y-intercept of each line to plot two points for each line on the same graph.points for each line on the same graph.

xx

yy Line #1: Line #1:

Slope = 1 and the y-int = -2Slope = 1 and the y-int = -2

Line #2:Line #2:

Slope = -2/3 and the y-int. = 3Slope = -2/3 and the y-int. = 3

The point of intersection of The point of intersection of the two lines is the point (3,1).the two lines is the point (3,1).

This system of equations has This system of equations has one solutionone solution, the , the pointpoint (3,1)(3,1)

xx

yyThe two equations in The two equations in slope-intercept form are:slope-intercept form are:y x

y x

3

2 6

Plot points for each line.Plot points for each line.

Draw in the lines.Draw in the lines.

This system of equations represents two intersecting lines.This system of equations represents two intersecting lines.

The solution to this system of equations is a single point The solution to this system of equations is a single point (3,0) . (3,0) .

System of Equations:“Solutions”

“No Solutions” “Infinite Solutions”

• when lines of a graph are when lines of a graph are parallelparallel

• since they do not intersect, since they do not intersect, there is there is no solutionno solution

• a pair of equations a pair of equations that have the same that have the same slope and y-intercept.slope and y-intercept.

•What is happening What is happening here??here??

xx

yy The two equations in The two equations in slope-intercept form are:slope-intercept form are:

y x

y x o r y x

1

31

3

9

9

9

1

31

Plot points for each line.Plot points for each line.Draw in the lines.Draw in the lines.

These two equations represent the same line.These two equations represent the same line.Therefore, this system of equations has Therefore, this system of equations has infinitely many solutions .infinitely many solutions .

With Your Buddy:On the graph paper I have given you: You and your elbow On the graph paper I have given you: You and your elbow partner graph and solve the system of linear equations. partner graph and solve the system of linear equations. Determine whether the following equations have one, none, or Determine whether the following equations have one, none, or infinite solutions. If “one solution” graph it and give the point infinite solutions. If “one solution” graph it and give the point of intersection. of intersection.

1)1) y = x - 1y = x - 1

y = 3y = 3

2)2)

ANS:ANS: One SolutionOne Solution (6,3)(6,3)

ANS:ANS: No Solution, the No Solution, the lines are parallellines are parallel

In-Class Practice 6-1

Determine whether the following have one, none, or infinite Determine whether the following have one, none, or infinite solutions by looking at the solutions by looking at the slopeslope and and y-interceptsy-intercepts, and graph , and graph each system.each system.

Closure

• Write a 4 sentence summary, in your own words, on Systems of Equations.– Remember, a summary is how you

would explain it to one of your friends if they asked you.