Solve the following equations by Substitution/ Elimination method
1. 11x + 15y + 23 = 0; 7x – 2y – 20 = 0.
2. 2x + y = 7; 4x – 3y + 1 = 0.
3. 23x – 29 y = 98; 29x – 23y = 110.
4. 2x + 5y = 8 / 3 ; 3x – 2y = 5 / 6 .
5. 4x – 3y = 8; 6x – y = 29 / 3 .
6. 2x - 3/4y = 3; 5x = 2y + 7.
7. 2x – 3y = 13; 7x – 2y = 20.
8. 3x – 5y – 19 = 0; –7x + 3y + 1 = 0.
9. 2x – 3y + 8 = 0; x – 4y + 7 = 0.
10. x + y = 5xy; 3x + 2y = 13xy.
11. 152 378 74; 378 152 604 x y x y .
12. 47 31 63;31 47 15 x y x y .
13. 71x + 37y = 253; 37x + 71y = 287.
14. 37x + 43y = 123; 43x + 37y = 117.
15. 217x + 131y = 913; 131x + 217y = 827.
16. 41x – 17y = 99; 17x – 41y = 75.
17. `\frac5x+6y=13; \frac3x+4y=7,\;x\neq 0`
18. `\frac2x+\frac3y=\frac9{xy}; \frac4x+\frac9y=\frac{21}{xy},\; x\ne 0, \;y\ne 0`
CONDITIONS FOR SOLVING LINEAR EQUATIONS
1. Find the value of k, so that the following system of equations has no solution: 3x – y – 5 = 0; 6x – 2y – k = 0.
2. Find the value of k, so that the following system of equations has a non-zero solution: 3x + 5y = 0; kx + 10y = 0.
Find the value of k, so that the following system of equations has no solution:
Find the value of k, so that the following system of equations has a unique solution:
For what value of k, the following pair of linear equations has infinite number of solutions:
Find the value of a and b for which each of the following systems of linear equations has a infinite number of solutions:
Show that the following system of the equations has a unique solution and hence find the solution of the given system of equations.
Solve each of the following system of linear equations graphically:
1. x+2y =3; 4x+3y = 2.
2. 2x+3y =8; x-2y+3 = 0.
3. x+2y+2 = 0; 3x+2y-2 = 0.
4. 4x+3y = 5; 2y- x = 7.
5. 2x-3y =1; 3x-4y =1.
6. 2x+3y = 4; 3x- y = -5.
7. x- y+1= 0; 3x+2y-12 = 0.
8. 3x+2y = 4; 2x-3y = 7.
9. 2x+3y = 2; x-2y =8.
10. 2x-5y+4 = 0; 2x+ y-8 = 0.
11. 3x+ y+1= 0; 2x-3y+8 = 0.
12. Solve the following system of linear equations graphically: 2x-3y-17 =0; 4x+ y-13= 0. Shade the region bounded by the above lines and x-axis.
13. Solve the following system of linear equations graphically: 2x+3y = 4; 3x- y = -5. Shade the region bounded by the above lines and y-axis.
14. Solve the following system of linear equations graphically: 4x- y = 4; 3x+2y =14.. Shade the region bounded by the above lines and y-axis.
15. Solve the following system of linear equations graphically:
x+2y =5; 2x-3y = -4.. Shade the region bounded by the above lines and y-axis.
16. Draw the graphs
of the equations 4x- y-8 = 0; 2x-3y+6 = 0. Also determine the vertices of the triangle formed by the lines and x-axis.
17. Solve the following system of linear equations graphically: 2x- y =1; x- y = -1, region bounded by the above lines and y-axis.
18. Solve the following system of linear equations graphically: 5x- y = 7; x- y+1= 0. the area bounded by these lines and y-axis.
19. Solve the following system of linear equations graphically: 4x-3y+4 = 0; 4x+3y-20 = 0 .
Calculate the area bounded by these lines and x-axis.
20. Solve the following system of linear equations graphically: 4x-5y-20 = 0; 3x+5y-15= 0. Find the coordinates of the vertices of the triangle formed by these lines and y-axis.
21. Solve the following system of linear equations graphically: 2x-5y+4 = 0; 2x+ y-8 = 0. Find the points where these lines meet the y-axis.
22. Solve the following system of linear equations graphically: 2x+ y -5 = 0; x+ y-3= 0. Find the points where these lines meet the y-axis.
23. Solve the following system of linear equations graphically: 4x-5y+16 = 0; 2x+ y-6 = 0. Find the coordinates of the vertices
of the triangle formed by these lines and y-axis.
24. Draw the graphs
of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular
region.
25. Solve the following system of linear equations graphically: 3x+ y-11= 0; x- y-1= 0. Shade the region bounded by these lines and the y-axis. Find the points where these lines cut the y-axis.
WORD PROBLEMS
I. NUMBER BASED QUESTIONS SIMPLE PROBLEMS
1. The sum of two numbers is 137 and their difference is 43. Find the numbers.
2. The sum of thrice the first and the second is 142 and four times the first exceeds the second by 138, then find the numbers.
3. Sum of two numbers is 50 and their difference is 10, then find the numbers.
4. The sum of twice the first and thrice the second is 92 and four times the first exceeds seven times the second by 2, then find the numbers.
5. The sum of two numbers is 1000 and the difference between their squares is 25600, then find the numbers.
6. The difference between two numbers is 14 and the difference between their squares is 448, then find the numbers.
7. The sum of two natural numbers is 8 and the sum of their reciprocals is 8/15 . Find the numbers.
TWO-DIGIT PROBLEMS
1. The sum of the digits of a two digit number is 12. The number obtained by interchanging the two digits exceeds the given number by 18. Find the number.
2. Seven times a two-digit number is equal to four times the number obtained by reversing the order of its digit. If the difference between the digits is 3, then find the number.
3. The sum of the digits of a two digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.
4. The sum of the digits of a two digit number is 9. If 27 is added to it, the digits of the numbers get reversed. Find the number.
5. The sum of a two-digit number and the number obtained by reversing the digits is 66. If the digits of the number differ by 2, find the number. How many such numbers are there?
6. A two-digit number is 4 more than 6 times the sum of its digit. If 18 is subtracted from the number, the digits are reversed. Find the number.
7. The sum of a two-digit number and the number obtained by reversing the digits is 99. If the digits differ by 3, find the number.
8. The sum of a two-digit number and the number formed by interchanging its digit is 110. If 10 is subtracted from the original number, the new number is 4 more than 5 times the sum of the digits of the original number. Find the original number.
9. A two-digit number is 3 more than 4 times the sum of its digit. If 18 is added to the number, the digits are reversed. Find the number.
10. The sum of the digits of a two digit number is 15. The number obtained by interchanging the two digits exceeds the given number by 9. Find the number.
FRACTION PROBLEMS
1. A fraction becomes 9 /11 , if 2 is added to both the numerator and the denominator. If 3 is added to both the numerator and the denominator it becomes 5 / 6 . Find the fraction.
2. The sum of numerator and denominator of a fraction is 12. If the denominator is increased by 3 then the fraction becomes 1/2 . Find the fraction.
3. If 1 is added to both the numerator and denominator of a given fraction, it becomes 4/5 . If however, 5 is subtracted from both the numerator and denominator, the fraction becomes 1/ 2 . Find the fraction.
4. In a given fraction, if the numerator is multiplied by 2 and the denominator is reduced by 5, we get 6 /5 . But if the numerator of the given fraction is increased by 8 and the denominator is doubled, we get 2/ 5 . Find the fraction.
5. The denominator of a fraction is greater than its numerator by 11. If 8 is added to both its numerator and denominator, it reduces to 1/3 . Find the fraction.
II. AGE RELATED QUESTIONS
1. Ten years hence, a man’s age will be twice the age of his son. Ten years ago, man was four times as old as his son. Find their present ages.
2. A man’s age is three times the sum of the ages of his two sons. After 5 years his age will be twice the sum of the ages of his two sons. Find the age of the man.
3. If twice the son’s age in years is added to the mother’s age, the sum is 70 years. But if twice the mother’s age is added to the son’s age, the sum is 95 years. Find the age of the mother and her son.
4. Five years ago Nuri was thrice old as Sonu. Ten years later, Nuri will be twice as old as Sonu. Find the present age of Nuri and Sonu.
5. The present age of a woman is 3 years more than three times the age of her daughter. Three years hence, the woman’s age will be 10 years more than twice the age of her daughter. Find their present ages.
6. Two years ago, a man was 5 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 the man and his son.
7. I am three times as old as my son. Five years later, I shall be two and a half times as old as my son. How old am I and how old is my son?
8. A and B are friends and their ages differ by 2 years. A’s father D is twice as old as A and B is twice as old as his sister C. The age of D and C differ by 40 years. Find the ages of A and B.
9. The ages of two friends Ani and Biju differ by 3 years. Ani’s father Dharam is twice as old as Ani and Biju is twice as old as his sister Cathy. The ages of Cathy and Dharam differ by 30 years. Find the ages of Ani and Biju.
10. Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob’s age was seven times that of his son. What are their present ages?
11. A father is three times as old as his son. In 12 years time, he will be twice as old as his son. Find their present ages.
III. SPEED, DISTANCE AND TIME RELATED QUESTIONS
1. A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km down-stream. Determine the speed of the stream and that of the boat in still water.
2. Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?
3. 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 4 hours. Find their speeds.
4. A train covered a certain distance at a uniform speed. If the train would have been 10 km/h faster, it would have taken 2 hours less than the scheduled time. And, if the train were slower by 10 km/h; it would have taken 3 hours more than the scheduled time. Find the distance covered by the train. 5
5. Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current.
6. Roohi travels 300 km to her home partly by train and partly by bus. She takes 4 hours if she travels 60 km by train and the remaining by bus. If she travels 100 km by train and the remaining by bus, she takes 10 minutes longer. Find the speed of the train and the bus separately.
7. A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km down-stream. Determine the speed of the stream and that of the boat in still water.
8. 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 the rest by car, he takes 18 minutes longer. Find the speed of the train and that of the car.
9. A boat covers 32 km upstream and 36 km downstream in 7 hours. In 9 hours, it can cover 40 km upstream and 48 km down-stream. Find the speed of the stream and that of the boat in still water.
10. Two places A and B are 120 km apart on a highway. A car starts from A and another from B at the same time. If the cars move in the same direction at different speeds, they meet in 6 hours. If they travel towards each other, they meet in 1 hours 12 minutes. Find the speeds of the two cars.
IV. GEOMETRICAL
FIGURES RELATED QUESTIONS
1. The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle.
2. The length of a room exceeds its breadth by 3 metres. If the length is increased by 3 metres and the breadth is decreased by 2 metres, the area remains the same. Find the length and the breadth of the room.
3. The area of a rectangle gets reduced by 8m2, if its length is reduced by 5m and breadth is
increased by 3m. If we increase the length by 3m and the breadth by 2m, the area increases by 74m2. Find the length and the breadth of the rectangle.
4. In a DABC, ÐC = 3ÐB = 2(ÐA + ÐB). Find the angles.
5. Find the four angles of a cyclic quadrilateral ABCD
in which ÐA = (2x – 1)0, ÐB = (y + 5)0, ÐC = (2y + 15)0 and ÐD = (4x – 7)0.
6. The area of a rectangle remains the same if the length is increased by 7m and the breadth is decreased by 3m. The area remains unaffected if the length is decreased by 7m and the breadth is
increased by 5m. Find the dimensions of the rectangle.
7. The larger of two supplementary angles
exceeds the smaller by 18 degrees. Find them.
8. In a DABC, ÐA = x0, ÐB = (3x – 2)0, ÐC = y0. Also, ÐC – ÐB = 90. Find the three angles.
9. Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.
10. ABCD is a cyclic quadrilateral. Find the angles of the cyclic quadrilateral.
