> SUR[ ؊bjbj .ΐΐVQ cccccwww8\twNDMMMMMMMOPRDMuc"MccoN'''ccM'M''IMCXwRJ"MN0NJ5S!5SDM5ScM'MM%N5S ): sSolve the following equations:
(1) x+y=3 (2) x-2y=5
2x+5y =12 2x+3y=10
(3) 3x+y+1=0 (4) 2x+y-3=0
2x-3y+8=0 2x-3y-7=0
x+y=4 (6) x-2y=6
x-y=2 3x-6y=0
(7) x+y=4 (8) 2x+3y=4
2x-3y=3 x-y+3=0
(9) 2x-3y+13=0 (10) 2x+3y+5=0
3x-2y+12=0 3x-2y-12=0
Show that following systems of the equation has infinitely many equations
(11) 2y=4x-6
2x=y+3
Show that each one of the following systems of equations is in consistent(i.e. no solution):
(15) 3x-5y=20 (16) 2y-x=9
6x-10y= - 40 6y-3x=21
(17) x-2y=6
3x-6y=0
(18) Determine the vertices of the triangle, the equations of whose sides are given below:
2y-x=8,5y-x=14 and y-2x=1
y=x, y=0and 3x+3y=10
Determine whether the system of equations x-2y=2, 4x-2y=5 is consistent or in-consistent.
Determine whether the following system of linear equations has a unique solution or not:
2y=4x-6,2x=y+3
Solve each of the following systems of linear equations. Also find the coordinates of the points where the lines meet axis of y.
2x-5y+4=0 (ii)3x+2y=12
2x+y-8=0 5x-2y=4
(iii) 2x+y -11=0 (iv) x+2y-7=0
x-y-1=0 2x-y-4=0
(v) 3x+y-5=0 (vi) 2x-y-5=0
2x-y-5=0 x-y-3=0
Determine the coordinate of the vertices of a triangle, the equations of whose sides are:
y=x, y=2x and y + x=6
y=x,3y=x, x + y=8
Solve the following system of linear equations
(i) 2x+3y=12 , x-y=1 (ii) 3x+2y-4=0 , 2x-3y-7=0
(iii) 3x+2y-11=0 , 2x-3y+10=0
Find the coordinates of the vertices of the triangle formed by the two
Straight lines and the y-axis.
Draw the graphs of x-y+1=0 and 3x+2y-12=0. Calculate the area bounded by these lines and x-axis.
Solve the system of linear equations:
4x-3y+4=0
4x+3y-20=0
Find the area bounded by these lines and x-axis.
Solve the following system of linear equations
3x+y-11=0 , x-y-1=0.
Solve each of the following systems of linear equations. Also, find the area of the region bounded by these lines and y-axis.
Solve each of the following systems of linear equations. Also, find the coordinates of the points where the lines meet the axis of x in each system.
2x+y=6 (ii) 2x-y=2
x-2y=-2 4x-y=8
(iii) x+2y=5 (iv) 2x+3y=8
2x-3y=-4 x-2y= -3
Solve the following system of linear equations :
4x-5y-20=0
3x5y-15=0
Determine the vertices of the triangle formed by the lines representing the above equations.
Solve the following system of equations:
3x-4y=7 (ii) 4x-y=4
5x+2y=3 3x+2y=14
(32) Solve the following pair of equations
x+3y=6
2x-3y=12
(33)Solve the following systems of equations by using the method of
substitution.
3x-5y= -1 (ii) x+2y=-1
x-y=-1 2x-3y=12
(34) Solve the following systems of equations by using the method of
substitution:
2x+3y=9 (ii) EMBED Equation.DSMT4
3x+4y=5 EMBED Equation.DSMT4
Solve the following system of equations by using the method of
Elimination by equating the coefficients:
3x+2y=11 (ii) 8x+5y=9
2x+3y=4 3x+2y=4
Solve the following system of equations by using the method of elimination by equating the coefficients:
EMBED Equation.DSMT4
EMBED Equation.DSMT4
Solve the following system of equations:
EMBED Equation.DSMT4 , EMBED Equation.DSMT4 where x `"0, y `"-1
(38) Solve EMBED Equation.DSMT4 , EMBED Equation.DSMT4 and hence find a for which y=ax-4;
(39) Solve EMBED Equation.DSMT4 , EMBED Equation.DSMT4 and hence find p if y=px-2.
(40) Solve: 3(2u+v)=7uv , 3(u +3v)=11uv.
(41) Solve the following system of equations:
(i) EMBED Equation.DSMT4 , EMBED Equation.DSMT4 (ii) EMBED Equation.DSMT4 , EMBED Equation.DSMT4 ,y `" 0
(iii) EMBED Equation.DSMT4 , EMBED Equation.DSMT4 (iv) x+2y= EMBED Equation.DSMT4 , 2x+y= EMBED Equation.DSMT4
If 2x+y=35 and 3x+3y=65 find the value of x/y.
Solve EMBED Equation.DSMT4 , EMBED Equation.DSMT4 .
Solve EMBED Equation.DSMT4 , EMBED Equation.DSMT4 .
Solve the following systems of equations by using the method of cross- multiplication;
x + y=7 , 5x+12y=7
Solve ax + by = a b , bx ay =a + b
Solve EMBED Equation.DSMT4 , ax-by = a - b
Solve the following system of equations in x and y
(a-b)x + (a +b)y = a -2ab b
(a +b)(x +y) = a + b
(49) Solve the following system of equations in x and y:
EMBED Equation.DSMT4
EMBED Equation.DSMT4 , where x, y `" 0.
(50) Solve the following system of equations in x and y.
ax+ by=1 , EMBED Equation.DSMT4 or EMBED Equation.DSMT4
(51) Solve each of the following systems of equations by the method of cross multiplication:
(i) x+2y +1 =0 (ii) 3x+2y+25=0 , 2x+y+10=0
2x-3y-12=0 (iii) 2x+y =35 , 3x+4y=65
(iv)2x y=6 , x y = 2
Solve 2 (ax by) + a + 4b = 0 , 2(bx + ay) +b 4a=0
Solve EMBED Equation.DSMT4 , EMBED Equation.DSMT4
Solve 6(ax+by) =3a+2b , 6(bx-ay) = 3b 2a.
EMBED Equation.DSMT4 , EMBED Equation.DSMT4 , x , y `" 0
mx ny = EMBED Equation.DSMT4 , x + y = 2m.
For each of the following systems of equations determine the value of k for which the given system of equations has a unique solution:
(i) x-ky=2 , 3x + 2y =-5 (ii) 2x-3y=1 , kx + 5y = 7
(iii) 2x + 3y 5 = 0 , kx 6y 8=0 (iv) 2x +ky =1 , 5x 7y = 5
( 58) For each of the following systems of equations determine the value of k
for which the given system of equations has infinitely many solutions.
(i) 5x+2y=k , 10x +4y = 3 (ii) (k -3)x + 3y =k , kx + ky = 12
(iii) kx +3y = k-3 , 12x +ky = k
(59) For each of the following system of equations determine the value of k
For which the given system has no solution:
(i) 3x 4y + 7=0 , kx + 3y 5=0 (ii) 2x ky + 3 = 0 , 3x + 2y 1 =0 k
(60) For what value of k , will the following system of equations have
infinitely many solutions?
2x+3y=4 , (k+2)x +6y=3k+2
(61) Determine the values of a and b for which the following system of linear equations has infinite solutions:
2x-(a-4)y=2b+1 , 4x-(a-1)y=5b-1.
For what value of k will the following system of linear equations has no solution?
3x+y=1 , (2k-1)x +(k-1)y = 2k + 1
Find the value of k for which the following system of linear equations has infinite solutions:
x +(k+1)y=5 , ( k+1)x + 9y = 8k -1
( 64) For what value of k , will the system of equations
x + 2y = 5 , 3x +ky 15 = 0 has (i) a unique solution? (ii) no solution?
(65) Find the values of ( , ( for which the following system of linear equations has infinite number of solutions:
2x+3y = 7 , 2(x + ( ( + ( )y=28 has(i) a unique solution? (ii) no solution?
(66) Determine the value of k so that the following linear equations have no solution.
(3k + 1) x + 3y 2 = 0, ( k + 1)x + ( k-2) y -5 = 0
In each of the following systems of equations determine whether the system has a unique solution , no solution or infinitely many solutions. In case there is a unique solution, find it:
(i) x-3y=3 , 3x-9y=2 (ii) 2x+ y=5 , 4x+2y=10
(iii) 3x- 5y = 20 , 6x 10y = 40 (iv) x 2y = 8 , 5x 10y =10
Find the value of k for which the following system of equation has a unique solution:
(i) kx+2y=5 , 3x+y=1 (ii) x-2y=3 , 3x +ky= 1
(iii) 4x 5 y = k , 2x 3y = 12, (iv) x + 2y = 3, 5x + ky + 7=0
Find the value of k for which the following system of equations have infinitely many solutions:
(i) 2x + 3y -5 = 0 , 6x + ky 15 = 0 (ii) 4x + 5y= 3,kx+15y=9
(iii) 8x+5y=9 , kx+20y = 18 (iv)2x-3y=7 , (k+2)x (2k+1)y=3(2k-1)
(70) Find the value of k for which the following system of equations have infinitely many solutions:
(i) 2x+3y=2 , (k+2)x +(2k+1)y=2(k-1)
(ii) x + ( k + 1 ) y=4 , (k+1)x+9y = 5k+2
(iii)kx+3y=2k+1 , 2(k+1)x +9y=7k+1
(iv)2x+(k-2)y=k , 6x +(2k-1)y=2k+5
(v) 2x + 3y=7 , (k+1)x +(2k -1)y=4k+1
(71) 2x +3y =k , (k 1) x + (k + 2)y= 3k
Find the value of k for which the following equations has no solution(20 25)
(72) kx -5y=2 , 6x + 2y = 7 , (73) x+2y=0 , 2x + ky =5
(74) 3x-4y+7=0 , kx+3y-5=0 (75) 2x-ky+3=0 , 3x+2y-1=0
(76) 2x+ky=11 , 5x-7y=5 (77) cx+3y=3 , 12x+cy=6
(78) For what value of k the following system of equations will be inconsistent?
4x+6y=11 , 2x+ky=7
(79) For what value of ( , the system of equations
(x+3y= ( -3,12x+(y = ( will have no solution?
(80) Find the value of k for which the system kx+2y=5 , 3x+y=1 has (i) a unique solution, and (ii) no solution.
(81) For what value of k , the following system of equations will represent the coincident lines?
x+2y+7=0 ,2x+ky+14=0
(82) Obtain the condition for the following system of linear equations to have unique solution. ax + by=c , lx +my =n.
(83) Determine the values of a and b so that the following system of linear equations has infinite number of solutions: 2x -3y=7 , (a +b)x (a +b 3)y = 4a +b (84) Find the values of p and q for which the following system of linear equations has infinite number of solutions:
2x +3y=7 , (p+q)x+(2p-q)y = 3(p+q+1)
(85) Find the values of a and b for which the following system of equations has infinitely many solutions:
(i)(2a-1)x -3y=5 , 3x+(b-2)y=3 (ii) 2x-(2a+5)y=5 , (2b+1)x -9y=15
(iii) (a -1)x+3y+4y=12 , 6x+(1-2b)y=6
(iv)3x+4y=12 ,(a+b)x+2(a-b)y=5a-1 (v) 2x+3y=7,(a-b)x+(a+b)y=3a+b-2
(86) 4 chairs and 3 tables cost Rs 2100 and 5 chairs and 2 tables cost Rs 1750. Find the cost of a chair and a table separately.
(87) 2 tables and 3 chairs together cost Rs 2000 whereas 3 tables and 2 chairs together cost Rs 2500. Find the total cost of 1 table and 5 chairs.
(88) A and B each have certain number of oranges. A says to B,if you give me 10 of your oranges, I will have twice the number of oranges left with you. B replies, if you give me 10 of your oranges, I will have the same number of oranges as left with you. Find the number of oranges with A and B separately.
(89) A man has only 20 paisa coins and 25 paisa coins in his purse. If he has 50 coins in all totaling Rs 11.25, how many coins of each kind does he have?
(90) 5 pens and 6 pencils together cost Rs 9 and 3 pens and 2 pencils cost Rs.5. Find the cost of 1 pen and 1 pencil.
(91) 7 audio cassettes and 3 video cassettes cost Rs 1110, while 5 audio cassettes and 4 video cassettes cost Rs 1350. Find the cost of an audio cassette and a video cassette.
(92) Reena has pens and pencils which together are 40 in number. If she has 5 more pencils and 5 less pens, then number of pencils would become 4 times the number of pens. Find the original number of pens and pencils.
(93) 3 bags and 4 pens together cost Rs 257 whereas 4 bags and 3 pens together cost Rs 324. Find the total cost 1 bag and 10 pens.
(94) 5 books and 7 pens together cost Rs 79 whereas 7 books and 5 pens together cost Rs 77. Find the total cost of 1 book and 2 pens.
(95) Sum of two numbers is 35 and their difference is 13. Find the numbers.
(96) In a two digit , the units digit is twice the tens digit. If 27 is added to the number, the digits interchange their places. Find the number
(97) In a two digit number, the tens digit is three times the units digit. When the number is decreased by 54, the digits are reversed. Find the number.
(98) The sum of the digits of a two digit number is 8 and the difference between the number and that formed by reversing the digits is 18. Find the number.
(99) The sum of a two digit number and the number formed by interchanging its digits is 110. If 10 is subtracted from the first number, the new number is 4 more than 5 times the sum of the digits in the first number. Find the first number.
(100) The sum of a two digit number and the number formed by interchanging the digit is 132. If 12 is added to the number, the new number becomes 5 times the sum of the digits. Find the number.
(101) The sum of a two digit number and the number obtained by reversing the order of its digits is 165. If the digits differ by 3, find the number.
(102) The sum of the digits of a two digit number is 15. The number obtained by reversing the order of the digits of the given number exceeds the given number by 9. Find the given number.
(103) The sum of a two digit number and the number obtained by reversing the order of its digits is 99. If the digits differ by 3, find the number.
(104) A two-digit number 3 more than 4 times the sum of its digits. If 18 is added to the number , the digits are reversed. Find the number.
(105) A two-digit number is 4 more than 6 times the sum of its digits. If 18 is subtracted from the number, the digits are reversed. Find the number.
(106) A two-digit number is such that the product of its digits is 20. If 9 is added to the number, the digits interchange their places. Find the number.
(107) A fractions becomes 4/5 , if 1 is added to both numerator and denominator. If, however, 5 is subtracted from both numerator and denominator, the fraction becomes1/2 . What is the fraction?
(108) The denominator of a fraction is 4 more than twice the numerator. When both the numerator and denominator are decreased by 6 , then the denominator becomes 12 times the numerator. Determine the fraction.
(109) The numerator of a fraction is 4 less than the denominator. If the numerator is decreased by 2 and denominator is increased by 1, then the denominator is eight times the numerator. Find the fraction.
(110) A fraction becomes 9/11 if 2 is added to both numerator and the denominator. If 3 is added to both the numerator and the denominator it becomes 5/6. Find the fraction.
(111) The sum of a numerator and denominator of a fraction is 18. If the denominator is increased by 2, the fraction reduces to 1/3. Find the fraction.
(112) The sum of the numerator and denominator of a fraction is 4 more than twice the numerator. If the numerator and denominator are increased by 3, they are in the ratio 2:3. Determine the fraction.
(113) The sum of the numerator and denominator of a fraction is 12. If the denominator is increased by 3, the fraction becomes 1/2. Find the fraction.
(114) Ten years ago, father was twelve times as old as his son and ten years hence, he will be twice as old as his son will be. Find their present ages.
(115) Five years hence, fathers age will be three times the age of his son. Five years ago, father was seven times as old as his son. Find their present ages.
(116) A father is three times as old as his son. After twelve years, his age will be twice as that of his son then. Find their present ages.
(117) Ten years later, A will be twice as old as B and five years ago , A was three times as old as B. What are the present ages of A and B?
(118) A is elder to B by 2 years. As father F is twice as old as A and B is twice as old as his sister S. If the ages of the father and sister differ by 40 years, find the ages of A.
(119) Six years hence a mans age will be three times the age of his son and three years ago he was nine times as old as his son. Find their present ages.
(120) The present age of a father is three years more than three times the age of the son. Three years hence fathers age will be 10 years more than twice the age of the son. Determine their present age.
(121) Fathers age is three times the sum of ages of his two children. After 5 years his age will be twice the sum of ages of two children. Find the age of father.
(122) Two years ago a father was five times as old as his son. Two years later, his age will be 8 more than three times the age of the son. Find the present ages of father and son.
(123) Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu?
(124) Points A and B are 90 km apart from each other on a highway. A car starts from A and another from B at the same time. If they go in the same direction they meet in 9 hours and if they go in opposite directions they meet in 9/7 hours. Find their speeds.
(125) Ved travels 600 km to his home partly by car. He takes 8 hours if he travels 120 km by train and the rest by car. He takes 20 minutes longer if he travels 200 km by train and the rest by car. Find the speed of the train and the car.
(126) A man travels 370 km partly by train and partly by car. If he covers 250 km by train and the rest by car, it takes him 4 hours. But , if he travels 130 km by train and rest by car, he takes 18 minutes longer. Find the speed of the train and that of the car.
(127) Points A and B are 70 km. a part on a highway. A car starts from A and another car starts from B simultaneously. If they travel in the same direction, they meet in 7 hours, but if they travel towards each other, they meet in one hour. Find the speed of the two cars.
(128) A sailor goes 8 km downstream in 40 minutes and returns in 1 hour. Determine the speed of the sailor in still water and the speed of the current.
(129) Ramesh travels 760 km to his home partly by train and partly by car. He takes 8 hours if he travels 160 km. by train and the rest by car. He takes 12 minutes more if the travels 240 km by train and the rest by car. Find the speed of the train and car respectively.
(130) A man travels 600 km partly by train and partly by car. If the covers 400 km by train and the rest by car, it takes him 6 hours and 30 minutes. But, if the travels 200 km by train and the rest by car, he takes half an hour longer. Find the speed of the train and that of the car.
(131) Places A and B are 80 km apart from each other on a highway. A car starts from A and other from B at the same time. If they move in the same direction, they meet in 8 hours and if they move in opposite directions, they meet in 1 hour and 20 minutes. Find the speeds of the cars.
(132) A boat goes 12 km upstream and 40 km downstream in 8 hours. It can go 16 km upstream and 32 km downstream in the same time. Find the speed of the boat in still water and the speed of the stream.
(133) Abdul traveled 300 km by train and 200 km by taxi, it took him 5 hours 30 minutes. But if he travels 260 km by train and 240 km by taxi he takes 6 minutes longer. Find the speed of the train and that of the taxi.
(134) The taxi charges in a city comprise of a fixed charge together with the charge for the distance covered. For a journey of 10 km the charge paid is Rs 75 and for a journey of 15 km the charge paid is Rs 110. What will a person have to pay for traveling a distance of 25 km?
(135) A person invested some amount at the rate of 12% simple interest and some other amount at the rate of 10% simple interest. He received yearly interest of Rs 130. But if he had interchanged the amounts invested, he would have received Rs 4 more as interest. How much amount did he invest at different rates?
(136) A man sold a chair and a table together for Rs 1520 thereby making a profit of 25% on the chair and 10% on table. By selling them together for Rs 1535 he would have made a profit of 10% on the chair and 25% on the table. Find the cost price of each.
(137) Students of a class are made to stand in rows. If one student is extra in a row, there would be 2 rows less. If one student is less in a row there would be 3 rows more. Find the number of students in the class.
(138) The ratio of incomes of two persons is 9:7 and the ratio of their expenditures is 4:3. If each of them saves Rs 200 per month find their monthly incomes.
(139) In a (ABC, ( C=3 ( B=2 ( ( A+( B). Find the three angles.
(140) The car hire charges in a city compromise of a fixed charges together with the charge for the distance covered. For a journey of 12 km, the charge paid is Rs 89 and for a journey of 20 km, the charge paid is Rs 145. What will a person have to pa!"BCopPQh𬤬ylj_ h+h+EHUjcxK
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(141) A part of monthly hostel charges in a college are fixed and the remaining depend on the number of days one has taken food in the mess. When a student A takes food for 20 days, he has to pay Rs 1000 as hostel charges whereas a student B, who takes food for 26 days, pays Rs 1180 as hostel charges. Find the fixed charge and the cost of food per day.
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Topic: - linear equation in two variables s
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