"1Pair Of Linear Equation In Two VariablesUnit bundle? Illustrations? Exercise? Do it yourself? Answer KeysLinear equations in Two variables Sample unit Outline Topic Page No.Day 1 Obtaining the linear equation in 2,3,4,5two variablesDay 2 Obtaining the linear equation in 6,7,8,9,10two variablesDay 3 Solving Linear Equations By 11,12,13,14,15Graphical MethodDay 4 Solving Linear Equations By 16,17,18,19,20Graphical MethodDay 5 Solving Linear Equations By 21,22,23,24Graphical MethodDay 6 Solving Linear Equations by 25,26,27,28,29,30Substitution MethodDay 7 Word Problem of Linear Equations 31,32,33,34,35,36,37by Substitution MethodDay 8 Solving Linear Equations by 38,39,40,41,42Elimination MethodDay 9 Solving Linear Equations by 43,44,45,46Elimination MethodDay 10 Word Problem of Linear Equations 47,48,49,50,51by Elimination MethodDay 11 Word Problem of Linear Equations 52,53,54,55,56by Elimination MethodDay 12 Solving Linear Equations by Cross 56,57,58,59,60multiplication MethodDay 13 Word Problem of Linear Equations 61,62,63,64,65,66by Cross multiplication MethodDay 14 Word Problem with Any method 66,67,68,69,70,71Day 15 Test2IllustrationFather tells his son, “five years ago, I was seven times as old as youwere. After five years, I will be three times as old as you were”.Form thelinear equation in two Variables.? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? '? '? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? '? ' ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ,? ? ? ? ? ? ? ? ? ? ? : ? - 5? ? ? ? ? ? ? ? : ? - 5? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Father Son? - 5 = 7 (? - 5)? - 5 = 7? - 35? - 5 + 35- 7? = 0? -? ? +? ? =? --------------(? )? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? :? + 5? ? ? ? ? ? ? ? :? + 5? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?Father Son? + 5 = 3 (? + 5)? + 5 = 3? + 15? + 5- 3? - 15 = 0? - 3? + 5- 15 = 0? -? ? -? ? =? --------------- (? ) 3Do it yourself1. Father tells his son, “ten years ago, I was six times asold as you were. After five years, I will be four timesas old as you were”. Form Linear equations In twovariables(Answer :? - 6? + 50 = 0------------(1)? - 4? - 15 = 0------------(2)2. Father tells his son, “fifteen years ago, I was seventimes as old as you were. After ten years, I will befour times as old as you were”. Form Linearequations In two variables(Answer :? - 7? + 50 = 0------------(1)? - 4? - 30 = 0------------(2) 4The sum of the cost of 1 kg apple and 1kg pine apple isRs. 150 and the cost of 1 kg apple is twice the cost of 1kg pine apple. (? +? =? ? ? ,? ? -? =? ) ? ? ? ? ? ? ? ? ? ? ? ? ? 1? ? ? ? ? ? ? ? ? ? ? ? ?'? ' ? ? ? ? ? ? .' ? ? ? ? ? ? ? ? ? ? ? ? ? 1? ? ? ? ? ? ? ? ? '? ? ? ? ? ? ? .The sum of the cost of 1 kg apple and 1kg pine apple is Rs. 150? +? = 150--------------(1)The cost of 1 kg apple is twice the cost of 1 kg pine apple.? ? ? ? ?? ? ? ? ? ? ? ? ?2? = ?? 2? -? = 0Do it yourself1. The sum of the cost of 1kg apple and 1kg pine- apple is Rs.150.Cost of 1kg apple is four times thecost of 1kg pine – apple. (? +? =? ? ? ,? ? -? =? ) 5Nitin got twice the marks obtainedby Nihar in the annualth exam of 10 . The sum of the marks obtained by them is135.' ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? '? ? ? ? ? ? .? ? ? ? ? ? ? ? ? ? ? ? ? ? ? '? ' ? ? ? ? ? .The sum of the marks obtained by them is 135? +? = 135--------- (1)Nitin got twice the marks obtainedby Nihar? = 2 ?? - 2? = 0----------(2)Do it yourself1. Nitin got thrice the marks obtainedby Nihar in theth annual exam of 10 . The sum of the marks obtainedby them is 150. (? +? = 150,? - 3? = 0)6Length of a rectangle is five less than the thrice of its breadth. Theperimeter of the rectangle is 110. ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? "? "? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? "? "The perimeter of the rectangle is 110. ? +? +? +? = 1102? + 2? = 1102(? +? = 55)? +? =? ? ------------(? )Length of a rectangle is five less than the thrice of its breadth. ? =?? = 3? - 5? - 3? + 5 = 0---------- 2 Do it yourself1. Length of a rectangle is ten less than the Four time of its breadth.The perimeter of the rectangle is 110. (? +? =? ? ,? +? ? +? ? =? ) 7The sum of the weights of a father and a son is 85 kg. The weight of the? son isof the weight of his father.? ' ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? '? ' ? ?' ? ? ? ? ? ? ? ? ? ? ? ? '? '? ?The sum of the weights of a father and a son is 85 kg? +? =? ? ------------(? )? The weight of the son isof the weight of his father? ? =?? =? ?? -? ? =? --------------(? )? =?? =?? ? =? ?? -? ? =? --------------(? )Do it yourselfThe sum of the weights of a father and a son is 85 kg. The weightof the son is half of the weight of his father.(? +? = 85 , ? - 2? = 0) 8In a cricket match Sachin Tendulkar score thrice the sehwag’sscore . Bothof them together make a total score of 200 Runs.? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? '? ' ? ? ? ? .? ? ? ? ? ? ? ? ? ? ? ? ? ? '? ' ? ? ? ?Both of them together make a total score of 200 Runs.? +? =? ? ? ----------(? )Sachin Tendulkar score thrice the sehwag’s scoreSachin sehwag? =? ?? -? ? =? ------------- (? )Do it yourselfIn a cricket match Sachin Tendulkar score twice the sehwag’sscore . Bothof them together make a total score of 300 Runs. (? +? =? ? ? ,? -? ? =? ) 9Rakesh purchase 15 pens and 20 pencils. He has to pay Rs.190for it. Thesum of the cost of a pen and pencil is Rs.11. Obtain a pair of linearequation from the given data. ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? .?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?The sum of the cost of a pen and pencil is Rs.11.? +? =? ? ------------- (? )? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? .? ? ?? ? ? +? ? ? =? ? ? --------- (? )Do it yourselfA shopekeeper sells 5 pants and 8 shirst for Rs.3100, the cost of a pair ofpant and a shirt is Rs.500.(? +? =? ? ? ,? ? +? ? =? ? ? ? ) 10In tossing a balanced coin , the probability of getting a head on its face istwice to the probbality of getting a tail on its face. The sum of both theprobabilities (head and tail) is 1.? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? '? 'And the probability of getting tail on the face of coin is '? 'The sum of both the probabilities (head and tail) is 1 ? +? =? ------------- (? )The probability of getting a head on its face is twice to the probability ofgetting a tail on its face ? =? ?? -? ? =? ------------(? ) 11Solve the following linear equation and show it on the graph.? + 2? = 5&3? + 5? = 13? ? ? ? ? ? ? = 1 ? ? ? ? ? ? ? = 2 ? + 2? = 5 ? = 5- 2? ? = 5- 2?? = 5- 2?? = 5- 2(1) ? = 5- 2(2) ? = 5- 2 ? = 5- 4 ? = 3 ? = 1? ? +? ? =? ? ? ? ? ? ? ? ? =? ? ? ? ? ? ? ? =?? ? -? ? ? ? -? ?? ? =? ? -? ?? = ? =? ? ? ? -? ? ? =? ? -? (? ) ? ? -? (? )? ? =? =? ?? ? -? ? ? ? -? ? ? =? = ? ?? -? ? ? =? = ? ? ? =? ? =-? ? 1 -4? 2 5? ? ?? ? ? 12 13? ? +? ? =? ? &3? +? ? =? ?? ? +? ? =? ?? ? ? ? ? ? ? =? ? ? ? ? ? ? ? =? ? ? -? ? ? ? -? ? ? ? =? ? -? ?? =? = ? ? ? ? -? ?? = ? ? -? (? ) ? ? -? (? ) ? ? =? = ? ?? ? -? ? ? -? ? ? =? = ? ?? -? ? =? = ? ?? =? ? =-? ? ? +? ? =? ? ? ? ? ? ? ? ? =? ? ? ? ? ? ? ? =?? ? -? ? ? ? -? ? ? ? =? ? -? ?? =? = ? ? ? ? -? ?? = ? ? -? (? ) ? ? ? =? ? -? (? ) ? = ? ? ? -? ? = ? ? -? ? ? ? = ? ? ? ? = ? ? ? =? ? =?? =?? ? ?? ? -?? ? ? 14? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? . 15? +? =?& 3? +? ? =? ?? +? =?? ? +? ? =? ?? =? -? ? (? +? =? )? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? +? =?? ? ? ? ? ? ? .? =? -?? ? ? ? ? ? ? =? ? ? ? ? ? ? ? =? ? ? ? ? ? ? ? =? ? =? -?? =? -? ? =? -? ? =? -?? =? -? ? =? -? ? =?? =? ? =? ? ? ? ?? ? ? ? 16? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? .17Do it yourself? .? ? +? =?& ? + 6? =? ? (Solution Will be 3,2) ? = 8- 2?? 0 1 ? 8 6? ? -? ? = ? ? ? -?? ? ?? . ? +? ? =?& 3? +? ? =? (? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? )? =? -?? ? ? ? ? ?? -? ? ? =? ? ? ?? ? ? ?? ? .? ? +? ? =? & ? +? = 2 (? ? ? ? ? ? ? ? ? ? 1,1)5- 2? ? = 3 ? ? ? ? 1 -?? = 2-?? ? ?? ? ? 18? .? -? =?&3? -? ? =? ? (? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? )? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? = -6 +?? ? -? ? -5 -?? ? ? ? ? ? ? ? ? ? ? ?Draw the graph of the linear equation 3? + 2? = 5and 2? - 3? =-1 Determine the co-ordinates of the vertices of thetriangle formed by these linear equation and ? axis.? ? ? ? ? ? ? =? ? ? ? ? ? ? ? =?3? + 2? = 5 5- 3? 5- 3? 2? = 5- 3?? =? = 2 25- 3? ? = 5- 3(1) 5- 3(3) 2 ? =? =2 25- 3 5- 9 ? =? = 2 22 -4 ? =? = 2 2? = 1 ? =-2 ? ? ?? ? -?? ? ?? ? ? 19? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? .? ? ? ? ? ? ? =? ? ? ? ? ? ? ? =?2? - 3? =-1 -1- 2? -1- 2? -3? =-1- 2?? = ? =-3 -3-1- 2? ? =-1- 2(1) -1- 2(4) -3 ? = ? = -3 -3-1- 2 -1- 8 ? =? = -3 -3-3 -9 ? =? = -3 -3? = 1 ? = 320Thus the co ordinates of the vertices of the triangle formed by these linear?equation and the X axis are? ,? ,?(? .? ,? )? Do it yourselfDraw the graph of the linear equation ? + 3? = 6and 3? - 3? = 12 Determine the co-ordinates of the vertices of thetriangle formed by these linear equation and ? axis.[Answer : (6,0)(0,-4)(0,2)]Draw the graph of the linear equation ? +? = 10and ? -? = 4 Determine the co-ordinates of the vertices of the triangleformed by these linear equation and ? axis. 21 [? ? ? ? ? ? : ? ,? ? ? ,? (? ,? )]? +? =?&5? +? ? =? ? (? ? ? ? ? ? ? ? ? ? ? ,? )? ? -? ? ? =? -? ? =? ? ? ? ? ? ?? ? ? ? ? ? ? ? +? ? =? & 3? +? ? =? ?? -? ? ? ? -? ?? = ? =? ? ? -? ? ? ? ?? ? ? ? ? ?? +? ? -? =? ? , ? -? ? +? =? ? ? +?? -? =? ?? ? -? +? ? -? =? ?? ? -? ? -? +? ? -? =?-? +? ? -? =?-? +? ? =?? ? =? +? 22? +? ? =? ? ? ? ? ? ? ? =? ? ? ? ? ? ? ? =? ? +? ? +? ? = ? =? ?? +? ? +? ? = ? =? ?? ? ? =? = ? ?? =? ? =?? -?? +? =? ?? ? +? -? -? =? ?? ? -? ? +? -? -? =?? -? -? =?-? =? -?? =-? +?? ? ? ? ? ? ? =? ? ? ? ? ? ? ? =-? ? =-? +? ? =-? +? ? =-? +? ? =-? + (-? ) ? =? ? =-? -? ? =-?? ? ? ? ? -? 23? ? ? ? ? -? Lines for both the equations are parallel so No solution15 students of class X took part in the examination of the Indianmathematic Olympiad. The number of boy participants is five lessthan the number of girls participants find the number of girls and boywho participated.? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? '? '? ? ? ? ? ? ? ? ? ? ? ? ? '? '? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 15?? +? = 15------------(1) 24The number of boy participants is five less than the number of girlsparticipants? ? ? ? ? ? ? ? ??? =? - 5?? -? =-5----------(2)? ??? + 5 =??? -? =-5----------(2)? ? ? ? ? ? ? =? ? ? ? ? ? ? ? =? ? +? = 15 ? = 15-? ? = 15-?? = 15-? ? = 15- 7 ? = 15- 5 ? = 8 ? = 10? ? ? ? ? ? ? =? ? ? ? ? ? ? ? =? ?? -? =-5 ?? = 5 +? ?? = 5 +??-? =-5-? ?? = 5 + 1 ?? = 5 + 2?? =-(-5-? ) ?? = 6 ?? = 7 ?? = 5 +?? ? ? ? ? ?? ? ? ? ? ? ?25Number of Boys = 5. & Number of girls =10.10 students of class X took part in a quiz. If number of girls is 4 morethan the number of boys, find the number of girls and boys who tookpart in the quiz. (Answer is (x,y) = (3,4) ) ? ? ? ? ? ?? ? ? ? ? ? Substitution Method 26Illustration? +? =? ?&? -? = 4? +? =? ? --------(? )? -? =? --------(? )? +? =? ?? =? ? -?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? -? =? ? ?? - ? ? -? =? ? =? ? -? ? -? =? ? -? ? +? =? ? =? ? -? ? -? =?? +? =? ? +? ? =? -? =? -? ? ? =? ? -? =-? ? ? ? =?? =?? =? Solution (x,y)=(9,5)? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Do it yourself? ? -? =?&? -? + 6 = 0? ,? = (? ,? )? ? ? ? ? ? ? ? ? ? ? ?? ? +? ? =? ?& 10? +? ? ? =? ?? ? +? ? =? ? --------(? )? ? ? +? ? ? =? ? ------- (? ) 27? (? ? +? ? =? ? )Not possible ? there is no solution for the given polynomial ? + 4? = 8 &2? + 8? = 16? + 4? = 8--------- (1)2? + 8? = 16------- -(2)2(? + 4? = 8)? ? 1 ? ? ? ? ? 2 ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?Do it yourself? +? =? &3? -? =? (2,5)? ? +? ? =?&2? +? ? =?(No solution)? -? =?&3? -? ? =? (Infinitely Many solutions)? +? =?& 5? +? ? ? =? (? ,? =? ? ,-? )Illustration5? - 3? = 1 , 2? + 5? = 19 285? - 3? = 1------------- (1)2? + 5? = 19------------- (2)2? + 5? = 19-31? =-62 5? - 3? = 1-62 1- 5? ? 2? + 5 = 19 ?? =-3? = 1- 5?-3 -311- 5? 5- 25??? = 2? =? 2? + = 19-3 -3 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2? ×-3 5- 25? × 1 ? ? ? ? 1 ? ? ? ? 2? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? +? ? ? ? ?1 ×-3 -3× 119 ×-3 = 1 ×-3 ?-6? + 5- 25? =-57?-6? - 25? =-57- 55? - 3? = 1 ?-3? =-9 -9 ? 5 2 - 3? = 1?? =-3 ? 10- 3? = 1?? = 3? ? ? ? ? ? ? ? ? ,? = (2,3)?-3? = 1- 10Do it yourself? ? -? ? =-? ? &4? -? ? +? ? =?(Infinitely many solutions)? ? ? -? +? =? &? = 2? -(? ? ? ? ? ? ? ? ? ? )?? ? +? +? =? &2? -? ? +? =? ? ,? = (-? ,? ) 29?? ? -? ? =?&34? -? ? =? ?? ,? = ,? ? ? +? ? =? ?&2? +? ? ? =? ? ? ,? = (? ,? )? +? ? ? =?&8? +? ? ? =?? ,?? ? ? ? ? = ,? ? ( , )? ? ? ? ? ? ? ? ? ? ? ? ? :? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? +? ? =?&3? +? ? =?? ,? = (-? ,? )? ? ? ? ? ? ? ? - =-? & + =? ? ? ? ?30? ? ? ? - =-? --------- (? )? ? ? ? ? ? + = ----------(? )? ? ? ? ? ? ? + =? ? ? ? ? ? ? - =-?? ? ? ×? ? ×? ? ? ×? ? + =? ?×? ? ?×? ? ×? ? ×? ?×? ? - ? ×? ? ×? -? ×? ?? ? +? ? =? ?=? ×? -? ? -? ? ?? ? +?=? ?-? ? ?? ? -? ? ? =-? ?-? ? -? ? ? ?? ? + =? ??-? ? ? =-? ? -? ?-? ? ? ? ×-? ?-? ? -? ? ? ?? = ?×-? ? -? ? -? ? -? ? ?×? + -? ?×? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ×-? ? =?×-? ? ?-? ? ? -? ? -? ? ? =-? ? ?-? ? ?-? ? ? -? ? -? ? ? =-? ? ??? =-? ? ?-? ? ? =-? ? ? +? ? 31 ?? =??-? ? ? =-? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?1 OR 2 ?-? ? ? =-? ? -? ? ?-? ? ? =-? ? -? ??? =-? ? ?-? ? ? =-? ? -? (? )?? =??-? ? ? =-? ? -? ?? ? ? ? ? ? ? ?? ,? = (? ,? )Solve the pair of linear equation ? -? =? ? ? ? ? ? -? ? =? ? ? ?if thesolution satisfies ? =? ? +? ,? ? ? ? ? ? ? ? ? ?? -? =? ? ----------- (? ) ? -? ? =? ----------(? )? -? =? ?-? =? ? -?? =-(? ? -? )? =-? ? +? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? -? ? =? ?-? ? =-? ? -? ? ?? -? -? ? +? =??? =-? ?? +? ? -? ? =??? =? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??? -? ? =-? ?? -? =? ? ?-? =-? ? ?? ? -? =? ? ?? =? ? ?-? =? ? -? ? ? ? ? ? ? ? ? ? ? ? ?? ,? = (? ? ,? ? ) 32? ? ? =? ? +? ? ? -? =? ? ?=? ? =? ? ? ? ? ? =? ? ? +? ? =? ? ??×?? =? ? =? ? ? +? ? ? ×? ? ? ? ? ? ? ? ?Solve the pair of linear equation ? ? +? ? =? ? ? ? ? ? ? -? ? =-? ? ? ? ?if the solution satisfies ? =? ? +? ,? ? ? ? ? ? ? ? ? ? ? ,? = -? ,? ,? =-?Solve the pair of linear equation ? ? +? =? ? ? ? ? ? +? =? ? ? ? ?if the?solution satisfies ? =? ? +? ,? ? ? ? ? ? ? ? ? ? ? ,? = ? ? ,? ? ,? =? Illustration? A fraction becomes ,? ? ? is added to both numerator and denominator.? ? ? If 3 is added to both numerator and denomoninator it becomes . Find the? fraction?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? '? '? ? ? ? ? ? ? ? ? ? ? ? ? ? '? '? A fraction becomes ,? ? ? is added to both numerator and denominator ? ? ? + 2 9 ? 11? + 22 = 9? + 18? =? + 2 11 ? 11? - 9? = 18- 22? + 2 × 11 9× (? + 2) ? =? 11? - 9? =-4--- (1)? + 2× 11 11 × (? + 2) ? If 3 is added to both numerator and denominator it becomes .?33? + 3 5 =? 6? - 5? = 15- 18? + 3 6 ? 6? - 5? =-3----- (2)(? + 3) × 6 5 × (? + 3) ? =? + 3× 6 6 × (? + 3) ? 6? + 18 = 5? + 15? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?11? - 9? =-4-9? =-4- 11?-4- 11? ? =-9 -(4 + 11? ) ? =-9 4 + 11? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ?9 6? - 5? =-3 ? 54? - (20 + 55? ) =-27 4 + 11? ? 54? - 20- 55? =-276? - 5=-39 ? 54? - 55? =-27 + 2020 + 55? ? 6? - =-3?-? =-7-9 ?? = 7? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 346?× 9 20 + 55?× 1 ? -1 × 9 -9× 1 -3× 9 =1 × 9 7 6? - 5? =-3 -5? =-3- 42? ? ? ? ? ? ? ? ? ? ? ? ? ?96 7 - 5? =-3 -5? =-45 -4542- 5? =-3? =-5 ? = 9? ? ? ? ? ? ? ? ? ? ? ?? Afraction becomesif 3 is added to the both the? numerator and denominator . If 5 is added to? Numerator and the denominatorit becomesfind? ? that fraction. ? ? ? ? ? ? = ? ? ? ? ? : 5? - 4? = -3, 6? - 5? = -5? Afraction becomesif 1 is added to the both the? numerator and denominator . If 5 is added to 35? Numerator and the denominatorit becomesfind? ? that fraction. ? ? ? ? ? ? = ? ? ? ? ? : 5? - 4? = -1,7? - 6? = -5? A fraction becomeswhen 2 is subtracted from the? ? numerator and denominator. It becomeswhen 5 is? added to its denominator and numerator, find the? fraction?? ? ? ? ? ? =? ? ? ? ?? ? . 1 = 5? - 2? = +6 ? ? . 2 = 4? - 3? = -5? ? ? ? ? ? ? ? ? ? ? ?The cost of a table is thrice the cost of a chair. The total cost of4 chairsand a table is Rs.2100. Find the cost of a table and cost of a chair. ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? .'? '? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? '? 'The total cost of4 chairs and a table is Rs.2100. ? +? ? =? ? ? ? ----------(? )The cost of a table is thrice the cost of a chair. ? ??= ? ? 36? -? ? =? ------(? )? +? ? =? ? ? ?? =? ? ? ? -? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ?? -? ? =? -? ? =-? ? ? ? -? ? ? ? ? ? ? ? -? ? -? ? =?? = -? ? ? ? ? -? ? =?? =? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? -? ? =? ?? -? ? ? ? =??? -? ? ? =??? =? ? ?Cost of a chair is Rs.300 & Cost of a table is Rs.900The sum of present ages of father and son is 50 years. After 5 years, the ageof father becomes thrice the age of his son. Find their present ages.' ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? '? ' ? ? ? ? ? ? ? ? ' ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? '? ' ? ? ? ? ?The sum of present ages of father and son is 50 years.? +? =? ? --------- (? )? ? ? ? ? 5 ? ? ? ? ?Father’s age? +?Son’s age ? +?After 5 years, the age of father becomes thrice the age of his son.37(? +? ) =? ( ? +? )?? +? =? ? +? ??? -? ? =? ? -??? -? ? =? ? ------- (? )Solve further by Substitution method Solution is ? ? ? ? ? ? =? ? ? ? ? ? ? :? ? ? =? ? ? ? ? ? ?Do it yourselfThe sum of present age of a father and twice the present ageof a son is 70 years. Also, the sum of twice the present age of afather and the present age of son is 95 years. Find theirpresent ages.? ? ? ? :? ? ? ? ? ? '? ' ? ? ? "? " (? +? ? = ? ? , ? ? +? = ? ? ) ? ,? = ? ? ,? ? ? ? ? ? ? ? ? ? ?A bus traveller travelling with some of his relatives buys 5tickets from ahmedabad to Anand and 10 tickets fromahmedabad to Vadodara for $1100. The total cost of oneticket from ahmedabad to Anand and one ticket fromahmedabad to Vadodara Is $140. Find the cost of a ticketfrom ahmedabad to Anand as well as the cost of a ticket fromAhmedabad to Vadodara. ? ? ? ? ? ? ? cost of one ticket from ahmedabad to Anand is Rs.'? ' ? ? ? ? ? ? ? ? ? one ticket from ahmedabad to Vadodara Is Rs.' ? ' 38A bus traveller travelling with some of his relatives buys 5 ticketsfrom ahmedabad to Anand and 10 tickets from ahmedabad toVadodara for $1100.5? + 10? = 1100----------(1)The total cost of one ticket from ahmedabad to Anand and one ticketfrom ahmedabad to Vadodara Is $140.? +? = 140--------- (2)5? + 10? = 1100 ? +? = 140 ?-5? =-4001100- 10? -400 ? 5? = 1100- 10?+? = 140 ?? =5 -5 1100- 10??? =1100- 10?× 1 ? × 5 ?? = 80$5 ? +5 × 1 1 × 5 140 × 5 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ?? ? 21 × 5 ? 1100- 10? + 5? = 700?-5? = 700- 1100? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? +? =? ? ? ? ? .?? +? ? =? ? ??? =? ? ? -? ??? =? ? $cost of one ticket from ahmedabad to Anand is $60cost of one ticket from ahmedabad to vadodara is $ 8039Do it yourselfA bus traveller travelling with some of his relativesbuys 5 tickets from ahmedabad to Anand and 10tickets from ahmedabad to Vadodara for $1050. Thetotal cost of one ticket from ahmedabad to Anand andone ticket from ahmedabad to Vadodara Is $130. Findthe cost of a ticket from ahmedabad to Anand as wellas the cost of a ticket from Ahmedabad to Vadodara. 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