Linear Equations in Two Variables


 Terence Perkins
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1 Liner Equtions in Two Vribles In this chpter, we ll use the geometry of lines to help us solve equtions. Liner equtions in two vribles If, b, ndr re rel numbers (nd if nd b re not both equl to 0) then x + by = r is clled liner eqution in two vribles. (The two vribles re the x nd the y.) The numbers nd b re clled the coe cients of the eqution x+by = r. The number r is clled the constnt of the eqution x + by = r. Exmples. 10x 3y = nd 2x 4y = 7 re liner equtions in two vribles. Solutions of equtions A solution of liner eqution in two vribles x+by = r is specific point in R 2 such tht when when the xcoordinte of the point is multiplied by, nd the ycoordinte of the point is multiplied by b, nd those two numbers re dded together, the nswer equls r. (There re lwys infinitely mny solutions to liner eqution in two vribles.) Exmple. Let s look t the eqution 2x 3y =7. Notice tht x =ndy =1ispointinR 2 tht is solution of this eqution becuse we cn let x =ndy =1intheeqution2x 3y =7 nd then we d hve 2() 3(1) = 10 3=7. The point x =8ndy =3islsosolutionoftheeqution2x 3y =7 since 2(8) 3(3) = 16 9=7. The point x =4ndy =6isnot solutionoftheeqution2x 3y =7 becuse 2(4) 3(6) = 8 18 = 10, nd 10 6= 7. To get geometric interprettion for wht the set of solutions of 2x 3y =7 looks like, we cn dd 3y, subtrct 7, nd divide by 3 to rewrite 2x 3y =7 s 2 3 x 7 3 = y. This is the eqution of line tht hs slope 2 3 nd yintercept 7 of 3. In prticulr, the set of solutions to 2x 3y =7isstrightline. (This is why it s clled liner eqution.) 244
2 .. Liner Liner equtions equtions nd lines. nd lines. If b =0,thenthelinerequtionx If b + by = + r by is = the r sme is the s sme x = s r. x = r. Dividing Dividing by gives by x gives = x = r of r, the solutions to this eqution consist, so the solutions to this eqution consist of the of the points on points the verticl on the verticl line whose line xcoordintes whose xcoordintes equl r. equl r. ~xtr (b* If b = 6= 0, If then b = 6= the 0, then sme the ides sme from ides the from 2x the 3y 2x =7exmplethtwelooked 3y t previously t previously shows tht shows x tht + by x = di erent r+ is by the = r sme is the eqution sme eqution s, just s, written just written in in di erent form from, form from, b x + r b = y. This is the eqution of stright line whose slope is b nd b x + r b = y. This is the eqution of stright line whose slope is whose yintercept b nd whose yintercept is r b. is r b. ~xtr (b*o) U 0 o oj 0 o H
3 Systems of liner equtions Rther thn sking for the set of solutions of single liner eqution in two vribles, we could tke two di erent liner equtions in two vribles nd sk for ll those points tht re solutions to both of the liner equtions. For exmple, the point x =4ndy =1issolutionofbothoftheequtions x + y =ndx y =3. If you hve more thn one liner eqution, it s clled system of liner equtions, so tht x + y = x y =3 is n exmple of system of two liner equtions in two vribles. There re two equtions, nd ech eqution hs the sme two vribles: x nd y. A solution of system of equtions is point tht is solution of ech of the equtions in the system. Exmple. The point x =3ndy =2issolutionofthesystemoftwo liner equtions in two vribles 8x +7y =38 3x y = 1 becuse x =3ndy =2issolutionof3x y = 1 nd it is solution of 8x +7y =38. Unique solutions Geometriclly, finding solution of system of two liner equtions in two vribles is the sme problem s finding point in R 2 tht lies on ech of the stright lines corresponding to the two liner equtions. Almost ll of the time, two di erent lines will intersect in single point, so in these cses, there will only be one point tht is solution to both equtions. Such point is clled the unique solution of the system of liner equtions. Exmple. Let s tke second look t the system of equtions 8x +7y =38 3x y = 1 246
4 The first eqution in this system, 8x +7y =38,correspondstolinetht The first8 eqution in this system, 8x +7y =38,correspondstolinetht hs slope 7 8. The second eqution in this system, 3x y =3,isrepresented hs slope by line tht hs slope 3 7. The second eqution 3 in this system, 3x y =3,isrepresented. Since the two slopes re not equl, the by line tht hs slope lines hve to intersect in exctly 3 = 3 one. Since the two slopes re not equl, the point. Tht one point will be the unique lines hve to intersect in exctly one point. Tht one point will be the unique solution. As we ve seen before, x =3ndy =2issolutionofthissystem. solution. As we ve seen before tht x =3ndy =2issolutiontothis It is the unique solution. system, it must be the unique solution. Ii ii ~ Exmple. The system Exmple. The system x +2y =4 S x +2y =4 2x =11 2x + y =11 hs unique solution. It s 2ndy =7. hs unique solution. It s x = 2ndy =7. It s strightforwrd to check tht 2ndy =7issolutionofthe It s strightforwrd to 3 check tht x = 2ndy =7issolutiontothe system. Tht it s the only solution follows from the fct tht the slope of the system. Tht it s the only solution follows from the fct tht the slope of the line x +2y =4isdi erent from slope of the line 2x =11. Thosetwo line x +2y =4isdi erent from slope of the line 2x + y =11. Thosetwo slopes re 2 nd 2 respectively. slopes re 2 nd 2 11 respectively. No No solutions solutions. If you rech into ht nd pull out S two di erent liner equtions in two If you rech into ht nd pull out two di erent liner equtions in two vribles, it s highly unlikely tht the two corresponding lines would hve vribles, it s highly unlikely tht the two corresponding lines would hve exctly the sme slope. But if they did hve the sme slope, then there exctly the sme slope. But if they did hve the sme slope, then there
5 Ii. would of of no wouldnot notbe be solution solutionto the system of two liner equtions since no point in R 2 3of to the system of two liner equtions since no point in R 2 would wouldlie lieon onboth bothof ofthe theprllel prllellines. lines. 2. Exmple. Exmple. The Thesystem x 2y = x 2y = 44 3x +6y =0 does not hve solution. ii 3x +6y ~ =0 Tht s becuse ech of of the two lines hs the sme does not slope, 1 hve solution. Tht s becuse ech of the two lines hs the sme slope, 2, 1 so the lines don t intersect. 2, so the lines don t intersect. S It 3 S * * * * * * * * * * * * * * * * * * * * * * * * * *
6 Exercises 1.) Wht re the coe cients of the eqution 2x y = 23? 2.) Wht is the constnt of the eqution 2x y = 23? 3.) Is x = 4ndy =3solutionoftheeqution2x y = 23? 4.) Wht re the coe cients of the eqution 7x +6y =1?.) Wht is the constnt of the eqution 7x +6y =1? 6.) Is x =3ndy = 10 solution of the eqution 7x +6y =1? 7.) Is x =1ndy =0solutionofthesystem x + y =1 2x +3y =3 8.) Is x = 1ndy =3solutionofthesystem 7x +2y = 1 x 3 y = 14 9.) Wht s the slope of the line 30x 6y =3? 10.) Wht s the slope of the line 10x +y =4? 11.) Is there unique solution to the system 30x 6y =3 10x +y =4 12.) Wht s the slope of the line 6x +2y =4? 13.) Wht s the slope of the line 1x +y = 7? 249
7 14.) Is there unique solution to the system 6x +2y =4 1x +y = 7 For #117, find the roots of the given qudrtic polynomils. 1.) 9x 2 12x ) 2x 2 3x ) 4x 2 +2x 3 20
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