Linear Equations in Two Variables

1 Linear Equations in Two VariablesIn this chapter, we ll use the geometry of lines to help us solve Equations in two variablesIfa,b,andrare real numbers (and ifaandbare not both equal to 0) thenax+by=ris called alinear equation in two Variables . (The two Variables are thexand they.)The numbersaandbare called thecoe cientsof the equationax+by= numberris called theconstantof the equationax+by= 3y=5and 2x 4y= 7 are Linear Equations in of equationsAsolutionof a Linear equation in two variablesax+by=ris a specific pointinR2such that when when thex-coordinate of the point is multiplied bya,and they-coordinate of the point is multiplied byb, and those two numbersare added together, the answer equalsr. (There are always infinitely manysolutions to a Linear equation in two Variables .)

2 S look at the equation 2x 3y= thatx=5andy=1isapointinR2that is a solution of thisequation because we can letx=5andy=1intheequation2x 3y=7and then we d have 2(5) 3(1) = 10 3= pointx=8andy=3isalsoasolutionoftheequati on2x 3y=7since 2(8) 3(3) = 16 9= pointx=4andy=6isnotasolutionoftheequatio n2x 3y=7because 2(4) 3(6) = 8 18 = 10, and 106= 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=7as23x 73=y. This is the equation of a line that has slope23and ay-interceptof 73. In particular, the set of solutions to 2x 3y=7isastraightline.(This is why it s called a Linear equation.)244 Linear Equations and linesIfb=0,thenthelinearequationax+by=ri s the same asax= byagivesx=ra, so the solutions of this equation consist of thepoints on the vertical line whosex-coordinates 0, then the same ideas from the 2x 3y=7examplethatwelookedat previously shows thatax+by=ris the same equation as, just written inadi erent form from, abx+rb=y.

3 This is the equation of a straight linewhose slope is aband whosey-intercept Equations and ,thenthelinearequationax+by=ris the same asax= byagivesx=ra, so the solutions to this equation consist of thepoints on the vertical line whosex-coordinates = 0, then the same ideas from the 2x 3y=7examplethatwelookedat previously shows thatax+by=ris the same equation as, just written inadi erent form from, abx+rb=y. This is the equation of a straight linewhose slope is aband whosey-intercept Equations and ,thenthelinearequationax+by=ris the same asax= byagivesx=ra, so the solutions to this equation consist of thepoints on the vertical line whosex-coordinates 0, then the same ideas from the 2x 3y=7examplethatwelookedat previously shows thatax+by=ris the same equation as, just written inadi erent form from, abx+rb=y.

4 This is the equation of a straight linewhose slope is aband whosey-intercept ~xtr(b*o). Linear Equations and ,thenthelinearequationax+by=ris the same asax= byagivesx=ra, so the solutions to this equation consist of thepoints on the vertical line whosex-coordinates = 0, then the same ideas from the 2x 3y=7examplethatwelookedat previously shows thatax+by=ris the same equation as, just written inadi erent form from, abx+rb=y. This is the equation of a straight linewhose slope is aband whosey-intercept Equations and ,thenthelinearequationax+by=ris the same asax= byagivesx=ra, so the solutions to this equation consist of thepoints on the vertical line whosex-coordinates 0, then the same ideas from the 2x 3y=7examplethatwelookedat previously shows thatax+by=ris the same equation as, just written inadi erent form from, abx+rb=y.

5 This is the equation of a straight linewhose slope is aband whosey-intercept ~xtr(b*o)0-oJ\j-RU 0-oojH0-oJ\j-RU 0-oojHSystems 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 tobothof the Linear example, the pointx=4andy=1isasolutionofbothoftheequa tionsx+y=5andx y= you have more than one Linear equation, it s called asystemof 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 of a systemof Equations is a point that is a solution of each ofthe Equations in the pointx=3andy=2isasolutionofthesystemoftw olinear Equations in two variables8x+7y=383x 5y= 1becausex=3andy=2isasolutionof3x 5y= 1andit is a solution of8x+7y= solutionsGeometrically, finding a solution of a system of two Linear Equations in twovariables is the same problem as finding a point inR2that lies on each of thestraight lines corresponding to the two Linear 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.

6 Such a point is called theunique solutionof the system of s take a second look at the system of equations8x+7y=383x 5y= 1246 The first equation in this system , 8x+7y=38,correspondstoalinethathas slope 87. The second equation in this system , 3x 5y=3,isrepresentedby a line that has slope 3 5=35. Since the two slopes are not equal, thelines have to intersect in exactly one point. That one point will be the uniquesolution. As we ve seen before,x=3andy= is the unique system5x+2y=4 2x+y=11has a unique solution. It sx= 2andy= s straightforward to check thatx= 2andy=7isasolutionofthesystem. That it s the only solution follows from the fact that the slope of theline 5x+2y=4isdi erent from slope of the line 2x+y=11.

7 Thosetwoslopes are 52and 2 solutionsIf you reach into a hat and pull out two di erent Linear Equations in twovariables, it s highly unlikely that the two corresponding lines would haveexactly the same slope. But if they did have the same slope, then there247 The first equation in this system , 8x+7y=38,correspondstoalinethathas slope 87. The second equation in this system , 3x 5y=3,isrepresentedby a line that has slope 3 5=35. Since the two slopes are not equal, thelines have to intersect in exactly one point. That one point will be the uniquesolution. As we ve seen before thatx=3andy=2isasolutiontothissystem, it must be the unique system5x+2y=4 2x+y=11has a unique solution. It sx= 2andy= s straightforward to check thatx= 2andy=7isasolutiontothesystem.

8 That it s the only solution follows from the fact that the slope of theline 5x+2y=4isdi erent from slope of the line 2x+y=11. Thosetwoslopes are you reach into a hat and pull out two di erent Linear Equations in twovariables, it s highly unlikely that the two corresponding lines would haveexactly the same slope. But if they did have the same slope, then there188 The first equation in this system , 8x+7y=38,correspondstoalinethathas slope 87. The second equation in this system , 3x 5y=3,isrepresentedby a line that has slope 3 5=35. Since the two slopes are not equal, thelines have to intersect in exactly one point. That one point will be the uniquesolution. As we ve seen before thatx=3andy=2isasolutiontothissystem, it must be the unique system5x+2y=4 2x+y=11has a unique solution.

9 It sx= 2andy= s straightforward to check thatx= 2andy=7isasolutiontothesystem. That it s the only solution follows from the fact that the slope of theline 5x+2y=4isdi erent from slope of the line 2x+y=11. Thosetwoslopes are you reach into a hat and pull out two di erent Linear Equations in twovariables, it s highly unlikely that the two corresponding lines would haveexactly the same slope. But if they did have the same slope, then ~SIt3aSwould not be a solution of the system of two Linear Equations since no pointinR2would lie on both of the parallel systemx 2y= 4 3x+6y=0does not have a solution. That s because each of the two lines has the sameslope,12, so the lines don t intersect.**248would not be a solution to the system of two Linear Equations since no pointinR2would lie on both of the parallel systemx 2y= 4 3x+6y=0does not have a solution.

10 That s because each of the two lines has the sameslope,12, so the lines don t intersect.**189would not be a solution to the system of two Linear Equations since no pointinR2would lie on both of the parallel systemx 2y= 4 3x+6y=0does not have a solution. That s because each of the two lines has the sameslope,12, so the lines don t intersect.** ~SIt3aSExercises1.) What are the coe cients of the equation 2x 5y= 23 ?2.) What is the constant of the equation 2x 5y= 23 ?3.) Isx= 4andy=3asolutionoftheequation2x 5y= 23 ?4.) What are the coe cients of the equation 7x+6y=15?5.) What is the constant of the equation 7x+6y=15?6.) Isx=3andy= 10 a solution of the equation 7x+6y=15?7.) Isx=1andy=0asolutionofthesystemx+y=12x+3 y=38.