Linear Equations In Two Variables - Math

Linear Equations in Two VariablesIn this chapter, we’ll use the geometry of lines to help us solve equations.Linear equations in two variablesIf a, b, and r are real numbers (and if a and b are not both equal to 0) thenax by r is called a linear equation in two variables. (The “two variables”are the x and the y.)The numbers a and b are called the coefficients of the equation ax by r.The number r is called the constant of the equation ax by r.Examples. 10xvariables.3y 5 and2x4y 7 are linear equations in twoSolutions of equationsA solution of a linear equation in two variables ax by r is a specific pointin R2 such that when when the x-coordinate of the point is multiplied by a,and the y-coordinate of the point is multiplied by b, and those two numbersare added together, the answer equals r. (There are always infinitely manysolutions to a linear equation in two variables.)Example. Let’s look at the equation 2x 3y 7.Notice that x 5 and y 1 is a point in R2 that is a solution ofequation because we can let x 5 and y 1 in the equation 2x 3yand then we’d have 2(5) 3(1) 10 3 7.The point x 8 and y 3 is also a solution of the equation 2x 3ysince 2(8) 3(3) 16 9 7.The point x 4 and y 6 is not a solution of the equation 2x 3ybecause 2(4) 3(6) 8 18 10, and 10 6 7.this 7 7 7To get a geometric interpretation for what the set of solutions of 2x 3y 7looks like, we can add 3y, subtract 7, and divide by 3 to rewrite 2x 3y 7as 23 x 73 y. This is the equation of a line that has slope 23 and a y-interceptof 73 . In particular, the set of solutions to 2x 3y 7 is a straight line.(This is why it’s called a linear equation.)244

.-o0H.Linearequationsand lines.linesLinearLinearequationsand lines.Linearequationsandlines.equationsandIf bthen 0,0,thentheequationlinear equationequationax by rrrsameis thetheassameasr.ax r.r.If b If0,thelinearax rrbyistheax bybyax is thesameassameax r.Ifbb 0,thenthelinearequationax by thesameasax r.thenthelinearisisasaxrrrrr ,solutionsDividingby aaxa givesgives so thethe solutionssolutionsof thisthis equationequationconsistof thetheDividingDividingby a gives sothetoconsistofto thisthis equationequationconsist consistof thethe ofDividingbygivesxx , solutionssotothisequationconsistofthebyaa, xaaa, so the solutions swhosex-coordinatesx-coordinatesequal rra.points onthe verticallinewhoseequalx-coordinatesequal inatespointsonlinewhoseaa(b* xtrIfthen6 0,0,thenthenthesameideasfromthe2x3y 777exampleexamplethatwelookedlookedIf b thesameideasfromthefrom2x the3y thatlooked6 0,IfIf3y2x 77 exampleexamplethat wewethatlooked6 0,thenthesameideasfromthe2x3y examplethatwelookedbbb thesameideas3yweatpreviouslypreviouslyshowsthatax isbythe tenwritteninat previouslyshows thataxthat by ax r equationas,writteninequationas, justjustas,writteninatpreviouslyshowsthataxby thesameequationas,justwritteninatshowsby isissamejustinararar aa xy. rThisdi erentform from,from, y.y.Thisequationis thethe equationequationof aaa straightstraightlinea differentform from,istheoflineis Thistheequationof aa straightstraightline linedifferentformfrom,x brbb y.Thistheequationofstraightlineaaa differentformisisofbxbab x bb bbrarrararwhoseslopeis whoseandy-interceptwhose y-intercepty-interceptis b.whose slopeis slopeisis bb. isiswhoseslopeandwhosewhoseisisb andbbb and whose y-interceptbb-oU’018622452186oj(b*o) xtr

Systems of linear equationsRather than asking for the set of solutions of a single linear equation in twovariables, we could take two di erent linear equations in two variables andask for all those points that are solutions to both of the linear equations.For example, the point x 4 and y 1 is a solution of both of the equationsx y 5 and x y 3.If you have more than one linear equation, it’s called a system of linearequations, so thatx y 5x y 3is an example of a system of two linear equations in two variables. There aretwo equations, and each equation has the same two variables: x and y.A solution of a system of equations is a point that is a solution of each ofthe equations in the system.Example. The point x 3 and y 2 is a solution of the system of twolinear equations in two variables8x 7y 383x5y 1because x 3 and y 2 is a solution of 3x8x 7y 38.5y 1 and it is a solution ofUnique solutionsGeometrically, finding a solution of a system of two linear equations in twovariables is the same problem as finding a point in R2 that lies on each of thestraight lines corresponding to the two linear equations.Almost all of the time, two di erent lines will intersect in a single point,so in these cases, there will only be one point that is a solution to bothequations. Such a point is called the unique solution of the system of linearequations.Example. Let’s take a second look at the system of equations8x 7y 383x5y 1246

Thein this system, 8x 7y 38, correspondsto aa linethatThe firstfirst equation8x 7y7y 38,38, correspondscorresponds toto a lineline thatthat8equation in this system, 8xhasslope88. The second equation in this system, 3x 5y 3, is represented7has slope 77. The second equationin thisthis system,system, 3x3x 5y5y 3,3, isis representedrepresented33byalinethathasslope353 533. Since the two slopes are not equal, theby a line that has slopeSince thethe twotwo slopesslopes areare notnot equal,equal, thethe55 55. pointwillbetheuniquelines have to intersect in exactly one point.point. ThatThat oneone pointpoint willwill bebe thethe uniqueuniquesolution.Aswe’veseenbefore,x 3andy 2isasolutionofthissystem.solution. As we’ve seen before that xx 33 andand yy 22 isis aa solutionsolution toto thisthisItistheuniquesolution.system, it must be the unique solution.solution.5Ii.32.ii Example.Example. TheThe systemsystem5x 445x 2y2y2y 42x y 112x yy 1111Ithasy 7.has aa uniqueunique solution.solution. It’sIt’s xx 22 andandand yy 7.7.It’sstraightforwardtocheckthatx 2and yy 7 is aa solutionof theIt’s straightforward to3 check that xx 22 andand y 77 isis a solutionsolution toto hefactthattheslopeofthesystem. That it’s the only solution followsfollows fromfrom thethe factfact thatthat thethe slopeslope ofof thetheline5x 2y 4isdi erentfromslopeoftheline2x y 11.Thosetwoline 5x 2y 5 4 is different from slopeoftheline2x y 11.Thosetwoslope of the line 2x y 11. Those twoslopesare55 and 222respectively.2slopes are 22 and 11respectively.11SNoNo solutionssolutions.a Stwo di erent linear equations in twoIfIf youyou reachreach intointo aa hathat andand pullpull outoutout twotwo differentdifferent linearlinear equationsequations inin espondinglineswouldhavevariables, it’s highly unlikely that thethe twotwo correspondingcorresponding lineslines wouldwould eslope,thenthereexactly the same slope. But if theydid have the same slope, then there247 did have the same slope, then there1884

oftheparallellines.would lie on both of the parallel lines.2.Example.ThesystemExample.ThesystemExample. The systemxx 2y 4x 2y2y 443x 3x 6y6y6y 0003xii doesnothaveaa �tintersect.slope,22 ,1 so the lines don’t intersect.slope, , so the lines don’t intersect.2SIt3aS** ** ** ** ** ** ** ** ** ** ** ** ***************2481895

Exercises1.) What are the coefficients of the equation 2x2.) What is the constant of the equation 2x3.) Is x 5y 4 and y 3 a solution of the equation 2x4.) What are the coefficients of the equation5.) What is the constant of the equation6.) Is x 3 and y x y 12x 3y 31 and y 3 a solution of the system9.) What’s the slope of the line 30x10.) What’s the slope of the line1146y 3 ?10x 5y 4 ?11.) Is there a unique solution to the system30x6y 310x 5y 412.) What’s the slope of the line 6x 2y 4 ?13.) What’s the slope of the line 15x 5y 24923 ?5y 23 ?7x 6y 15 ?10 a solution of the equation7x 2y 5x3y 23 ?7x 6y 15 ?7.) Is x 1 and y 0 a solution of the system8.) Is x 5y 7?7x 6y 15 ?

14.) Is there a unique solution to the system6x 2y 415x 5y 7For #15-17, find the roots of the given quadratic polynomials.15.) 9x212x 416.) 2x23x 117.)4x2 2x3250

(The “two variables” are the x and the y.) The numbers a and b are called the coecients of the equation ax by r. The number r is called the constant of the equation ax by r. Examples. 10x 3y 5and2x 4y 7 are linear equations in two variables. Solutions of equations A solution of a linear equation in two variables ax by r is a .