PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Two liner equtions in the sme two vriles re lled pir of liner equtions in two vriles. The most generl form of pir of liner equtions is x + y + 0 x + y + 0 where,,,,,, re rel numers, suh tht + 0, + 0. A pir of liner equtions in two vriles n e represented, nd solved, y the: (i) grphil method (ii) lgeri method. Grphil Method : The grph of pir of liner equtions in two vriles is represented y two lines. (i) If the lines interset t point, then tht point gives the unique solution of the two equtions. In this se, the pir of equtions is onsistent. (ii) If the lines oinide, then there re infinitely mny solutions eh point on the line eing solution. In this se, the pir of equtions is dependent (onsistent). (iii) If the lines re prllel, then the pir of equtions hs no solution. In this se, the pir of equtions is inonsistent. 4. Algeri Methods : We hve disussed the following methods for finding the solution(s) of pir of liner equtions : (i) Sustitution Method (ii) Elimintion Method (iii) Cross-multiplition Method. If pir of liner equtions is given y x + y + 0 nd x + y + 0, then the following situtions n rise : (i) In this se, the pir of liner equtions is onsistent. (ii) In this se, the pir of liner equtions is inonsistent. www.exellup.om 009 send your queries to enquiry@exellup.om
(iii) In this se, the pir of liner eqution is dependent nd onsistent.. There re severl situtions whih n e mthemtilly represented y two equtions tht re not liner to strt with. But we lter them so tht they re redued to pir of liner equtions. EXERCISE. Aft tells his dughter, Seven yers go, I ws seven times s old s you were then. Also, three yers from now, I shll e three times s old s you will e. Represent this sitution lgerilly nd grphilly. Answer: Let us ssume tht seven yers go Aft s ge ws x yers nd his dughter s ge ws y. So, s per the question x 7y x 7 y 0 --------------------------------- (i) The eqution gives us the following tle: X Y 7 4 8 4 4 49 7 Now, three yers from now mens 0 yers from 7 yers k Aft ge will e x + 0 Dughter s ge will e y + 0 As per question x + 0 ( y + 0) x + 0 y + 0 x y + 0 x y 0 0 ---------------------------- (ii) www.exellup.om 009 send your queries to enquiry@exellup.om
The eqution gives us following tle: X Y 9 4 8 4 7 0 Fther's Age 0 40 0 0 0 0 x y 0 0 x 7 y 0 4 7 Dughter's Age. The oh of riket tem uys ts nd lls for Rs 900. Lter, she uys nother t nd more lls of the sme kind for Rs 00. Represent this sitution lgerilly nd geometrilly. Answer: Let us ssume numer of ts x nd tht of lls y. As per the question we get following equtions: x + y 900 x 900 y x 00 y ---------------------- (i) x + y 00 x 00 y ---------------------- (ii) As oth equtions re similr so there oinident lines in the grph, mening there n e infinite numer of solutions to this prolem. www.exellup.om 009 send your queries to enquiry@exellup.om
. The ost of kg of pples nd kg of grpes on dy ws found to e Rs 0. After month, the ost of 4 kg of pples nd kg of grpes is Rs 00. Represent the sitution lgerilly nd geometrilly. Answer: Let us ssume tht ost of pple x nd tht of grpes y As per the question we get following equtions: x + y 0 y 0 x --------------------------- (i) The eqution gives us following tle: X Y 8 4 4 0 4 x + y 00 y 00 4x y 0 x --------------------------- (ii) The eqution gives us following tle: X Y 48 4 44 4 4 40 www.exellup.om 009 send your queries to enquiry@exellup.om
0 Grpes 0 4 40 y0-x y0-x 0 4 Apples Sine we get prllel lines in this grph so there will e no solution s equtions re inonsistent. EXERCISE. Form the pir of liner equtions in the following prolems, nd find their solutions grphilly. (i) 0 students of Clss X took prt in Mthemtis quiz. If the numer of girls is 4 more thn the numer of oys, find the numer of oys nd girls who took prt in the quiz. Answer: Let us ssume numer of girls to e x nd tht of oys to e y. Then we get following equtions: x y + 4 -------------------- (i) The eqution will give following tle: X Y 4 0 7 8 4 9 x + y 0 x 0 y --------------- (ii) www.exellup.om 009 send your queries to enquiry@exellup.om
The eqution will give following tle: X Y 0 0 9 8 7 4 0 Girls 8 4 xy+4 x0-y 0 4 Boys As oth lines interset t 7, so Numer of Girls 7 nd tht of Boys (ii) penils nd 7 pens together ost Rs 0, wheres 7 penils nd pens together ost Rs 4. Find the ost of one penil nd tht of one pen. Answer: x + 7y 0 x 0 7y ------------------ (i) The eqution will give following tle: y X - -0 4-7 0 0 0-4 - 8 -. 9 -. 0-4 www.exellup.om 009 send your queries to enquiry@exellup.om
7 x + y 4 7x 4 y ----------------- (ii) The eqution gives following tle y X -9-8 - 9-7 Pen 0 0 0 0 - -0-7 9 7 9 7 9 Penil x0-7y 7x4-y Both lines re interseting t, so x Putting the vlue of x in either of the equtions we n get the vlue of y, whih is equl to. Also, is the only vlue of y whih gives similr vlues for x in oth the tles. So ost of one penil Rs. nd one pen Chek: x + 7y 0 + 7 + 0. On ompring the rtios, nd, find out if the lines representing the following pirs of liner equtions interset t point, re prllel or oinident: (i) x 4y + 8 0 7x + y 9 0 www.exellup.om 009 send your queries to enquiry@exellup.om
7 7 x x 4 4 y y 9 8 9 8 As it is ler tht, so lines will e interseting. (ii) 9x + y + 0 8x + y + 4 0 8 9 4 It is ler tht, so lines re oinidentl. (iii) x y + 0 0 x y + 9 0 9 0 It is ler tht, so lines will e prllel www.exellup.om 009 send your queries to enquiry@exellup.om
. On ompring the rtios, nd find out whether the following pir of liner equtions re onsistent, or inonsistent. (i) x + y ; x y 7 Here,, so the pir of liner equtions is onsistent. (ii) x y 8 ; 4x y 9 4 9 8 9 8 Here,, so the pir of liner equtions is inonsistent. (iii) 7 + y x, 9x 0y 4 9 0 Here,, so the pir of liner equtions is onsistent. www.exellup.om 009 send your queries to enquiry@exellup.om
(iv) x y ; 0x + y 0 Here,, so the pir of liner equtions is dependent nd onsistent. (v) 8 4 + y x ; x + y 4 8 In this se,, so the pir of liner eqution is dependent nd onsistent. 4. Whih of the following pirs of liner equtions re onsistent/inonsistent? If onsistent, otin the solution grphilly: (i) x + y, x + y 0 0 www.exellup.om 009 send your queries to enquiry@exellup.om
Both equtions will give the sme tle s follows: y x x' - 0 0-4 9 9-8 8-7 7-0 4 4 4 0 0 - - 7 - - 8 - - 9-4 -4 0 8 y 4 0 - -4-4 7 8 9 0 4 x x+y x+y0 As the lines re oinident so there n e infinitely mny solutions. In this se,, so the pir of liner eqution is dependent nd onsistent. www.exellup.om 009 send your queries to enquiry@exellup.om
(ii) x y 8, x y 8 Here,, so the pir of liner equtions is inonsistent. (iii) x + y 0, 4x y 4 0 4 Here,, so the pir of liner equtions is inonsistent. (iv) x y 0, 4x 4y 0 4 Here,, so the pir of liner equtions is inonsistent. www.exellup.om 009 send your queries to enquiry@exellup.om
. Hlf the perimeter of retngulr grden, whose length is 4 m more thn its width, is m. Find the dimensions of the grden. Answer: Let us ssume Length x nd redth y Then xy+4 Perimeter (x+y) Pr imeter x + y x + x + 4 x + 4 x 4 x Hene, y-4 So, Length m nd Bredth m. Given the liner eqution x + y 8 0, write nother liner eqution in two vriles suh tht the geometril representtion of the pir so formed is: (i) interseting lines (ii) prllel lines (iii) oinident lines Answer: (i) As you know tht if, then the lines will e interseting, so let us put numer for nd in suh wy whih stisfies this ondition. Next possile eqution n e x + y 7 0 (ii) As you know tht if eqution n e s follows: 4 x + y 8 0 (iii) As you know tht if, then the lines will e prllel, so the possile, then the lines will e oinidentl, so the possile eqution n e s follows: 4 x + y 0 www.exellup.om 009 send your queries to enquiry@exellup.om
EXERCISE. Solve the following pir of liner equtions y the sustitution method. (i) x + y 4, x-y4 Answer: x + y 4 x 4 y Sustituting this vlue of x in the seond eqution we get 4 y y 4 4 y 4 y y x 8 s (ii) s t, + t Answer: s +t Sustituting the vlue of s in the seond eqution we get + t t + + t + t + t t 0 t Hene, s +9 (iii) x y, 9x y 9 Answer: yx- Putting this vlue of y in the seond eqution we get 9x-(x-)9 9 x 9x + 9 9 It is ler tht this eqution will hve infinitely mny solutions. It is importnt to note tht here www.exellup.om 009 send your queries to enquiry@exellup.om
(iv) 0.x + 0.y. 0.4x + 0.y. Answer: x+y (fter multiplying the eqution y 0) y-x x Or, y Putting this vlue of y in the seond eqution we get x 4 x + (eqution is lso multiplied like eqution ) x + 0x 9 x + 9 x 4 x 4 y (v) x + y 0, x 8y 0 Answer: x y x y Putting this vlue of x in eqution we get y y y 8y 0 y 8y 0 0 y 4y 0 y 4y 0 y 4y Sine oeffiients of y re different on oth LHS nd RHS, so only possile vlue of y0 Hene, x0. Solve x + y nd x 4y 4 nd hene find the vlue of m for whih y mx +. Answer: x y Sustituting in eqution we get y 4y 4 7y 4 7 y + 4 y x 4 x www.exellup.om 009 send your queries to enquiry@exellup.om
Now putting vlues of x nd y in eqution we get m + m m. Form the pir of liner equtions for the following prolems nd find their solution y sustitution method. (i) The differene etween two numers is nd one numer is three times the other. Find them. Answer: Let us ssume one of the numers x nd nother numer y Then s per question x y ---------------- (i) x y -------------------------------------------- (ii) Sustituting the vlue of x in eqution (i) y y y y x 9 (ii) The lrger of two supplementry ngles exeeds the smller y 8 degrees. Find them. x + y 80 ------------- (i) x y 8 -------------- (ii) x y + 8 Sustituting in eqution (i) we get y + y + 8 80 y 80 8 y 8 x 8 + 8 99 (iii) The oh of riket tem uys 7 ts nd lls for Rs 800. Lter, she uys ts nd lls for Rs 70. Find the ost of eh t nd eh ll. Answer: 7 x + y 800 ----------- (i) x + y 70 --------------------- (ii) x 70 y 70 y x Sustituting in eqution (i) we get 7(70 y) + y 800 www.exellup.om 009 send your queries to enquiry@exellup.om
0 y + 8y 400 0 7y 400 7 y 0 400 80 y 0 70 0 x 00 Hene, Cost of Bt Rs. 00 Cost of Bll Rs. 0 (iv) The txi hrges in ity onsist of fixed hrge together with the hrge for the distne overed. For distne of 0 km, the hrge pid is Rs 0 nd for journey of km, the hrge pid is Rs. Wht re the fixed hrges nd the hrge per km? How muh does person hve to py for trvelling distne of km? Answer: Let us ssume the fixed hrge x And vrile hrge y As per question: x + 0 y 0 ------------------- (i) x + y ------------------- (ii) x y Sustituting in eqution (i) we get y + 0y 0 y 0 y 0 0 y 0 x 0 So for km the fir + 0 (v) A frtion eomes 9, if is dded to oth the numertor nd the denomintor. If, is dded to oth the numertor nd the denomintor it eomes. Find the frtion. Answer: Let us ssume the numertor x nd denomintor y As per question: x + 9 y + x + 9y + 8 9 y x 4 --------------- (i) x + y + x + 8 y + www.exellup.om 009 send your queries to enquiry@exellup.om
y x --------------- (ii) y x + y x + Sustituting in eqution (i) we get 4x + 7 x 4 4 x + 7 x 0 7 x 0 x 7 y 7 + 9 7 Hene, required frtion 9 (vi) Five yers hene, the ge of Jo will e three times tht of his son. Five yers go, Jo s ge ws seven times tht of his son. Wht re their present ges? Answer: Let us ssume five yers go son s ge x nd Jo s ge y Then y 7x ------------- (i) Son s ge five yers from now x+0 nd Jo s ge will e y+0 So, y + 0 ( x + 0) x + 0 y x + 0 -------------- (ii) Sustituting in eqution (i) x + 0 7x 4 x 0 x y Present ge of Jo 40 yers Present ge of son 0 yers www.exellup.om 009 send your queries to enquiry@exellup.om
EXERCISE 4. Solve the following pir of liner equtions y the elimintion method nd the sustitution method: (i) x + y nd x y 4 Answer: Multiplying eqution (i) y nd sutrting eqution (ii) from this s follows: x + y 0 x y 4 0 + y y, sustituting in eqution (i) x + 9 x (ii) x + 4y 0 nd x y Answer: Multiplying eqution (i) y nd eqution (ii) y nd sutrting ltter from former s follows: x + 8y 0 x y 0 + 4y 4 y, sustituting in eqution (i) x + 4 0 x x (iii) x y 4 0 nd 9x y + 7 Answer: Multiplying eqution (i) y nd sutrting eqution (ii) s follows: 9x y 4 0 9x y 7 0 0 0y + 0 0 y y, sustituting in eqution (i) 0 x 4 0 x 8 0 www.exellup.om 009 send your queries to enquiry@exellup.om
x x x y (iv) + nd x y Answer: Simplify eqution (i) x y + x + 4y x + 4y ----------------- (iii) Simplify eqution (ii) x y x y 9 ------------------- (iv) Sutrting eqution (iv) from (iii) s follows: x + 4y x y 9 0 + y y, sustituting in eqution (iii) x x x. Form the pir of liner equtions in the following prolems, nd find their solutions (if they exist) y the elimintion method : (i) If we dd to the numertor nd sutrt from the denomintor, frtion redues to. It eomes, if we only dd to the denomintor. Wht is the frtion? x + Answer: y + x + y + x y ------------ (i) x y + www.exellup.om 009 send your queries to enquiry@exellup.om
x y + ------------ (ii) Sustituting vlue of x from eqution (i) in eqution (ii) y y + y x (ii) The sum of the digits of two-digit numer is 9. Also, nine times this numer is twie the numer otined y reversing the order of the digits. Find the numer. Answer: Let two digits of the numer re x nd y. x + y 9 x 9 y ---------------- (i) Then First numer 0 x + y ( x is t 0s ple nd y is t unit s ple) Reversing the numer we get numer whih n e written s 0y + x As per question: 9(0 x + y) (0 y + x) 90 x + 9y 0y + x 90(9 y) + 9y 0y + (9 y) 80 90y + 9y 0y + 8 y 80 8y 8y + 8 99 y 80 8 79 y 79 99 8 x 9 8 Smller numer 8 nd lrger numer 8 8 9 8 (iii) Meen went to nk to withdrw Rs 000. She sked the shier to give her Rs 0 nd Rs 00 notes only. Meen got notes in ll. Find how mny notes of Rs 0 nd Rs 00 she reeived. Answer: x + y x y ------------ (i) 0 x + 00y 000 0 ( y) + 00y 000 0 0y + 00y 000 0 y 000 0 70 y Numer of hundred rupees notes x 0 Numer of fifty rupees notes. www.exellup.om 009 send your queries to enquiry@exellup.om