Graphing Linear Equations - NJCTL

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Algebra IGraphing Linear Equations2015-11-04www.njctl.org2

Table of ContentsLinear EquationsClick on the topic to go to that sectionGraphing Linear Equations Using InterceptsHorizontal and Vertical LinesPoint-Slope FormSlope-Intercept FormProportional RelationshipsTeacher NotesSlope of a LineVoinbolinofwSolving Linear EquationsScatter Plots and the Line of Best FitPARCC Sample QuestionsGlossary and Standards3

Linear EquationsReturn to Tableof Contents4

Linear EquationsAny equation must have at least one variable.Linear equations have either one or two variables and mayalso have a constant.The variables in a linear equation are not raised to any power(beyond one): they are not squared, cubed, etc.The standard form of a linear equation isAx By CWhere: x and y are variables A and B are coefficients and C is a constant A, B, and C are integers A 05

Linear EquationsThere are an infinite number of solutions to alinear equation.In general, each solution is an ordered pair ofnumbers representing the values for thevariables that make the equation true.For each value of one variable, the value ofthe other variable is determined.6

Linear EquationsThe fact that the solutions of linear equations are aninfinite set of ordered pairs helped lead to the idea thatthose could be treated as points on a graph, and thatthose points would then form a line.Which is why these are called "Linear Equations."The idea of merging algebra and geometry led toanalytic geometry in the mid 1600's.7

Graphing EquationsAnalytic Geometry A powerful combination ofalgebra and geometry. Independently developed,and published in 1637, byRene Descartes and Pierrede Fermat in France. The Cartesian Plane isnamed for Descartes.8

How would youdescribe to someonethe location of thesefive points so theycould draw them onanother piece ofpaper without seeingyour drawing?Math PracticeGraphing EquationsDiscuss.9

Graphing EquationsAdding this Cartesiancoordinate plane makesthat task simple sincethe location of eachpoint can be given byjust two numbers:an x- and y-coordinate,written as theordered pair (x, y)y105-10-50510x-5-1010

Graphing EquationsWith the Cartesian Planeproviding a graphicaldescription of locationson the plane, solutions ofequations (as orderedpairs) can be analyzedusing algebra.y105-10-50510x-5-1011

Graphing EquationsThe Cartesian Plane isformed by the intersectionof the x-axis and y-axis,which are perpendicular.It's also called aCoordinate Plane or anXY Plane.The x-axis is horizontal(side-to-side) and they-axis is a vertical(up and down).The axes intersect at theorigin.y105-10-50510x-5-1012

Graphing EquationsAn ordered pairrepresents asolution to a linearequation, and apoint on the plane.y10(4, 8)5The numbersrepresent the x- andy- coordinates: (x,y).The point (4, 8) isshown.-10-50510x-5-1013

Graphing EquationsA linear equation has aninfinite set of solutions.y10Graphing the pairs of xand y values whichsatisfy a linear equationforms a line (hence thename "linear" equation).5-10-50510x-5-1014

Graphing EquationsOne way to graph theline that represents thesolutions to a linearequation is to use atable to find a few setsof solutions.Since a line is uniquelydefined by any twopoints, finding three ormore points providesthe line, and a check tomake sure the pointsare correct.y105-10-50510x-5-1015

Graphing EquationsLet's graph the line:yy 2x 310We'll make a table, picksome x-values and thencalculate the matchingy-values to create orderedpairs to graph.We can pick any values forx, but will choose them sothat the resulting points: Are easy to plot. Are far enough apart toallow us to draw anaccurate line.5-10-50510x-5-1016

Graphing Equationsy 2x 3x1230-3yyWhile we only need twopoints to determine theline, it's good to checkwith some extra points.Use the equation to fillin the y-values in thetable.105-10-50510x-5-1017

Graphing Equationsy 2x 3x1230-3yy5793-3These are just a fewpoints on the line.105-10-5There are an infinitenumber of orderedpairs that satisfy theequation.0510x-5-10Let's draw the line that represents theinfinite set of solutions to this equation.18

Graphing Equationsy 2x 3yThe arrows on both endsof the line indicate that itcontinues forever in bothdirections.Because it is a line, itincludes an infinity ofpoints representing all thereal numbers.105-10-50510x-5-1019

Graphing Equationsyy - 1 x 93x-3-10 2 6y10click9.3333.click9click8.3333.click7clickNote: When the coefficientof your x term is a fraction,it's helpful to select pointsthat are multiples of thedenominator. In thisexample, -3, 0 and 6 hady-valuesthatintegers.Note: clicktowerereveal105-10-50510x-5-10Click on the points that are integers & the line to graph20

1 Given the equation, y 2x - 5, what is y when x 0?A -7C -3D 0AnswerB -521

2 Given an equation of y 2x - 5, what is y if x 1 ?2A -5C -1D 7AnswerB -422

3 Is (3, -5) on the line y 2x - 12 ?A yesC not enough informationAnswerB no23

4 Which point is on the line 4y - 2x 0 ?A (-2, 1)C (-2, -1)D (1, 2)AnswerB (0, 1)24

5 Which point lies on the line whose equation is 2x - 3y 9 ?A (0, 3)C (-3, 0)D (6, 1)AnswerB (-3, 1)25

6 Point (k, -3) lies on the line whose equation is x - 2y -2.What is the value of k?B -6C 6AnswerA -8D 826

Graphing Linear EquationsUsing InterceptsReturn to Tableof Contents27

x- and y-interceptsyTo graph a line, two pointsare required. One techniqueuses the x- and y- intercepts.The x-intercept is where agraph of an equation passesthrough the x-axis. Thecoordinates of the x-interceptare (a, 0), where a is anyreal number.The x-intercept of the linearequation shown is (2, 0)105-10-50510x-5-1028

x- and y-interceptsTo graph a line, two points arerequired. One technique usesthe x- and y- intercepts.The y-intercept is where agraph of an equation passesthrough the y-axis. Thecoordinates of the y-interceptare (0, b), where b is any realnumber.The y-intercept of the linearequation shown is (0, -2)y105-10-50510x-5-1029

7 What is the y-intercept of this line?yAnswer105-10-50510x-5-1030

8 What is the x-intercept of this line?yAnswer105-10-50510x-5-1031

9 What is the y-intercept of this line?yAnswer105-10-50510x-5-1032

10 What is the y-intercept of this line?yAnswer105-10-50510x-5-1033

11 What is the x-intercept of this line?y5-10-50510xAnswer10-5-1034

12 What is the x-intercept of this line?yAnswer105-10-50510x-5-1035

Graphing Linear Equations Using InterceptsThe technique of using intercepts works wellwhen an equation is written in Standard Form.Recall that a linear equation written in standardform is Ax By C, where A, B, and C areintegers and A 0.36

Graphing Linear Equations Using InterceptsExample: Find the x- and y-intercepts in the equation3x 5y 15. Then graph the equation.x-intercept: Let y 0:y-intercept: Let x 0:3x 5(0) 153x 0 153x 15x 5 so x-intercept is (5, 0)3(0) 5y 150 5y 155y 15y 3 so y-intercept is (0, 3)37

Graphing Linear Equations Using InterceptsExample: Find the x- and yintercepts in the equation3x 5y 15. Then graph theequation.y10x-intercept is (5, 0)5Clicky-intercept is (0, 3)ClickClick on the points & the linein the coordinate plane toreveal.-10-50510x-5-1038

Graphing Linear Equations Using InterceptsExample: Find the x- and y-intercepts in the equation4x - 3y 12. Then graph the equation.x-intercept: Let y 0:4x - 3(0) 124x 12x 3 so x-intercept is (3, 0)y-intercept: Let x 0:4(0) - 3y 12-3y 12y -4 so y-intercept is (0, -4)slide down39

Graphing Linear Equations Using InterceptsExample: Find the x- andy-intercepts in the equation4x - 3y 12. Then graphthe equation.y10x-intercept is (3, 0)5Clicky-intercept is (0, -4)ClickClick on the points & theline in the coordinate planeto reveal.-10-50510x-5-1040

Does anyone see a shortcut to finding the x- and yintercepts? How could your shortcut make the problemeasier?Math PracticeGraphing Linear Equations Using Intercepts41

Graphing Linear Equations Using InterceptsGiven the equation 4x - 3y 12, another way to look atthe intercept method is called the "cover-up method."If y 0, we can cover -3y up (because zero timesanything is 0) and solve the remaining equation.4x - 3y 12press -3ythat leaves us with 4x 12Clicksolve for xthe x-intercept is (3, 0)Click42

Graphing Linear Equations Using InterceptsIf x 0, we can cover that up and solve the remaining equation.press 4x4x - 3y 12leaves us with -3y 12Clicksolve for ythe y -intercept is (0, -4)Click43

Graphing Linear Equations Using InterceptsTry This:yFind the x- and yintercepts of y 3x - 9.Then graph the equation.5(3, 0)-10-50510xAnswer10-5Click on the points & theline in the coordinateplane to reveal.-10(0, -9)44

Answer113 Given the equation y x - 7, what is the x-intercept?245

Answer114 Given the equation y x - 7, what is the y-intercept?246

Answer15 Given the equation y - 3 4(x 2), what is the x intercept?47

Answer16 Given the equation y - 3 4(x 2), what is y-intercept?48

17 Given the equation x 3y 3, what is the y-intercept?A (3, 0)C (0, 4)D (0, 3)AnswerB (0, 1)49

18 Given the equation x 3y 3, what is the x-intercept?A (3, 0)C (0, 4)D (0, 3)AnswerB (0, 1)50

Horizontal and Vertical LinesReturn to Tableof Contents51

Horizontal & Vertical Linesy10A vertical line goes "up anddown".Select random points oneach line shown to the left.What are the similaritiesand differences betweenthe points on the verticallines?Discuss!Math PracticeHorizontal and vertical linesare different from slantedlines in the coordinateplane.5-10-50510x-5-1052

Horizontal & Vertical LinesNotice that each point onthe line furthest to the left allhave x-coordinates of -7.Examples of points on thisline are (-7, 2), (-7, 0),(-7, -3), etc.The same holds true for thepoints on all of the verticallines that follow. What is thecommon x-coordinateshared on the remaininglines?2nd line from the left: -33rd line from the left: 24th line from the left 8y105-10-50510x-5-10slide to view53

Horizontal & Vertical LinesA vertical line has theequation x a, where ais the x-intercept and thecommon x-coordinateshared by all of thepoints on the line.yx -7 x -3 x 210x 85Notice no y is containedin the equation.-10-50510x-5-1054

Horizontal & Vertical LinesA horizontal line goes"sideways".ySelect random points oneach line shown to the left.What are the similarities anddifferences between thepoints on the horizontal lines.Discuss!Math Practice105-10-50510x-5-1055

Horizontal & Vertical LinesNotice that each point on thetop line have y-coordinatesof 10. Examples of points onthis line are (-5, 10), (-2, 10),(0, -10), etc.The same holds true for thepoints on all of the horizontallines that follow. What is thecommon y-coordinate sharedon the remaining lines?2nd line from the top: 63rd line from the top: -24th line from the top: -5slide to viewy105-10-50510x-5-1056

Horizontal & Vertical LinesA horizontal line has theequation y b, where b isthe y-intercept and thecommon y-coordinateshared by all of the points

Nov 04, 2015 · 2 Algebra I Graphing Linear Equations 2015-11-04 www.njctl.org. 3 Linear Equations Graphing Linear Equations Using Intercepts Horizontal and Vertical Lines Slope of a Line Point-Slope Form Slope-Intercept Form Solving Linear Equations Scatter Plots and the Line of Best Fit Proportional Relationships