R.S. Aggarwal solutions for Mathematics [English] Class 10 chapter 3 - Linear Equations in Two Variables [Latest edition]

Solve the system of equations graphically:

2x + 3y = 2, x – 2y = 8

Solve the following system of equations graphically:

3x + 2y = 12, x – y + 1= 0

Solve the system of equations graphically:

2x + 3y = 8, x – 2y + 3 = 0

Solve the system of equations graphically:

2x – 5y + 4 = 0, 2x + y – 8 = 0

Solve the system of equations graphically:

3x + 2y = 12, 5x – 2y = 4

Solve the system of equations graphically:

3x + y + 1 = 0, 2x – 3y + 8 = 0

Solve the system of equations graphically:

2x + 3y + 5 = 0, 3x – 2y – 12 = 0

Solve the system of equations graphically:

2x – 3y + 13 = 0, 3x – 2y + 12 = 0

Solve the following system of equations graphically:

2x + 3y – 4 = 0, 3x – y + 5 = 0

Solve the system of equations graphically:

x + 2y + 2 = 0, 3x + 2y – 2 = 0

Solve the following given system of equations graphically and find the vertices and area of the triangle formed by these lines and the x-axis:

x – y + 3 = 0, 2x + 3y – 4 = 0

Solve graphically the system of equations

2x – 3y + 4 = 0

x + 2y – 5 = 0.

Find the coordinates of the vertices of the triangle formed by these two lines and the x-axis.

Solve the following given system of equations graphically and find the vertices and area of the triangle formed by these lines and the x-axis:

4x – 3y + 4 = 0, 4x + 3y – 20 = 0

Solve the following given system of equations graphically and find the vertices and area of the triangle formed by these lines and the x-axis:

x – y + 1 = 0, 3x + 2y – 12 = 0

Solve the following given system of equations graphically and find the vertices and area of the triangle formed by these lines and the x-axis:

x – 2y + 2 = 0, 2x + y – 6 = 0

Solve graphically the system of equations

2x – 3y + 6 = 0

2x + 3y – 18 = 0.

Find the coordinates of the vertices of the triangle formed by these two lines and the y-axis.

Solve graphically the system of equations

4x – y – 4 = 0

3x + 2y – 14 = 0.

Find the coordinates of the vertices of the triangle formed by these two lines and the y-axis.

Solve graphically the system of equations

x – y – 5 = 0

3x + 5y – 15 = 0.

Find the coordinates of the vertices of the triangle formed by these two lines and the y-axis.

Solve graphically the system of equations

2x – 5y + 4 = 0

2x + y – 8 = 0.

Find the coordinates of the vertices of the triangle formed by these two lines and the y-axis.

Solve graphically the system of equations

5x – y = 7

x – y + 1 = 0.

Find the coordinates of the vertices of the triangle formed by these two lines and the y-axis.

Solve graphically the system of equations

2x – 3y = 12

x + 3y = 6.

Find the coordinates of the vertices of the triangle formed by these two lines and the y-axis.

Show graphically that the system of equations 2x + 3y = 6, 4x + 6y = 12 has infinitely many solutions. 

Show graphically that the system of equations 3x – y = 5, 6x – 2y = 10 has infinitely many solutions.

Show graphically that the system of equations 2x + y = 6, 6x + 3y = 18 has infinitely many solutions.

Show graphically that the system of equations x – 2y = 5, 3x – 6y = 15 has infinitely many solutions.

Show graphically that the system of equations x – 2y = 6, 3x – 6y = 0 is inconsistent.

 Show graphically that the system of equations 2x + 3y = 4, 4x + 6y = 12 is inconsistent.

Show graphically that the system of equations 2x + y = 6, 6x + 3y = 20 is inconsistent.

Draw the graphs for the following equations on the same graph paper:

2x + y = 2

2x + y = 6

Find the co-ordinates of the vertices of the trapezium formed by these lines. Also, find the area of the trapezium so formed.

Solve for x and y:

x + y = 3, 4x – 3y = 26

Solve for x and y:

x – y = 3, `x/3 + y/2 = 6`

Solve for x and y:

2x + 3y = 0, 3x + 4y = 5

Solve for x and y:

2x – 3y = 13, 7x – 2y = 20

Solve for x and y:

3x – 5y – 19 = 0, –7x + 3y + 1 = 0

Solve for x and y:

3x – 5y = 4, 2y + 7 = 9x

Solve for x and y:

`x/2 - y/9 = 6, x/7 + y/3 = 5`

Solve for x and y:

`x/3 + y/4 = 11, (5x)/6 - y/3 = -7`

Solve for x and y:

`4x - 3y = 8, 6x - y = 29/3`

Solve for x and y: 

`2x - (3y)/4 = 3, 5x = 2y + 7`

Solve for x and y: 

`2x + 5y = 8/3, 3x - 2y = 5/6`

Solve for x and y: 

`2x + 3y + 1 = 0, (7 - 4x)/3 = y`

Solve for x and y:

0.4x + 0.3y = 1.7, 0.7x – 0.2y = 0.8

Solve for x and y:

0.3x + 0.5y = 0.5, 0.5x + 0.7y = 0.74

Solve for x and y:

7(y + 3) – 2(x + 2) = 14, 4(y – 2) + 3(x – 3) = 2

Solve for x and y:

6x + 5y = 7x + 3y + 1 = 2(x + 6y – 1)

Solve for x and y:

`(x + y - 8)/2 = (x + 2y - 14)/3 = (3x + y - 12)/11`

Solve for x and y: 

`5/x + 6y = 13, 3/x + 4y = 7 (x ≠ 0)`

Solve for x and y:

`x + 6/y = 6, 3x - 8/y = 5 (y ≠ 0)`

Solve for x and y:

`2x - 3/y = 9, 3x + 7/y = 2 (y ≠ 0)`

Solve for x and y:

`3/x + 8/y = -1, 1/x - 2/y = 2  (x ≠ 0, y ≠ 0)`

Solve for x and y:

`9/x - 4/y = 8, (13)/x + 7/y = 101 (x ≠ 0, y ≠ 0)`

Solve for x and y:

`5/x - 3/y = 1, 3/(2x )+ 2/(3y) = 5 (x ≠ 0, y ≠ 0)`

Solve for x and y:

`1/(2x) + 1/(3y) = 2, 1/(3x) + 1/(2y) = 13/6 (x ≠ 0, y ≠ 0)`

Solve for x and y:

4x + 6y = 3xy, 8x + 9y = 5xy (x ≠ 0, y ≠ 0)

Solve for x and y:

x + y = 5xy, 3x + 2y = 13xy (x ≠ 0, y ≠ 0)

Solve for x and y:

`5/(x + y) - 2/(x - y) = -1, 15/(x + y) + 7/(x - y) = 10`

Solve for x and y:

`3/(x + y) + 2/(x - y) = 2, 9/(x + y) - 4/(x - y) = 1`

Solve for x and y:

`5/(x + 1) - 2/(y - 1) = 1/2, 10/(x + 1) + 2/(y - 1) = 5/2`, x ≠ –1 and y ≠ 1

Solve for x and y:

`44/(x + y) + 30/(x - y) = 10, 55/(x + y) + 40/(x - y) = 13`

Solve for x and y:

`10/(x + y) + 2/(x - y) = 4, 15/(x + y) - 9/(x - y) = -2`, where x ≠ y, x ≠ –y

Solve for x and y:

71x + 37y = 253, 37x + 71y = 287

Solve for x and y:

217x + 131y = 913, 131x + 217y = 827

Solve for x and y:

23x – 29y = 98, 29x – 23y = 110

Solve for x and y:

`(2x + 5y)/(xy) = 6, (4x - 5y)/(xy) = -3`

Solve for x and y:

`1/(3x + y) + 1/(3x - y) = 3/4, 1/(2(3x + y)) - 1/(2(3x - y)) = -1/8`

Solve for x and y:

`1/(2(x + 2y)) + 5/(3(3x - 2y)) = (-3)/2, 5/(4(x + 2y)) - 3/(5(3x - 2y)) = 61/60`

Solve for x and y:

`2/((3x + 2y)) + 3/((3x - 2y)) = 17/5, 5/((3x + 2y)) + 1/((3x - 2y)) = 2`

Solve for x and y:

3(2x + y) = 7xy, 3(x + 3y) = 11xy (x ≠ 0 and y ≠ 0)

Solve for x and y:

x + y = a + b, ax – by = a² – b²

Solve for x and y:

`x/a + y/b = 2, ax - by = (a^2 - b^2)`

Solve for x and y:

px + qy = p – q, qx – py = p + q

Solve for x and y:

`x/a - y/b = 0, ax + by = a^2 + b^2`

Solve for x and y:

6(ax + by) = 3a + 2b, 6(bx – ay) = 3b – 2a

Solve for x and y:

ax – by = a² + b², x + y = 2a

Solve for x and y:

`(bx)/a - (ay)/b + a + b = 0, bx - ay + 2ab = 0`

Solve for x and y:

`(bx)/a + (ay)/b = a^2 + b^2, x + y = 2ab`

Solve for x and y:

x + y = a + b, ax – by = a² – b²

Solve for x and y:

a²x + b²y = c², b²x + a²y = d²

Solve for x and y:

`x/a + y/b = a + b, x/(a^2) + y/(b^2) = 2`

Solve for x and y:

If 2x + y = 23 and 4x – y = 19, find the values of (5y – 2x) and `(y/x - 2)`.

Solve the system of equations by using the method of cross multiplication:

x + 2y + 1 = 0, 2x – 3y – 12 = 0

Solve the system of equations by using the method of cross multiplication:

3x – 2y + 3 = 0, 4x + 3y – 47 = 0

Solve the system of equations by using the method of cross multiplication:

6x – 5y – 16 = 0, 7x – 13y + 10 = 0

Solve the system of equations by using the method of cross multiplication:

3x + 2y + 25 = 0, 2x + y + 10 = 0

Solve the system of equations by using the method of cross multiplication:

2x + 5y – 1 = 0, 2x + 3y – 3 = 0

Solve the system of equations by using the method of cross multiplication:

2x + y – 35 = 0, 3x + 4y – 65 = 0

Solve the following system of equations by using the method of cross multiplication:

7x – 2y = 3, 22x – 3y = 16

Solve the system of equations by using the method of cross multiplication:

`x/6 + y/15 - 4 = 0, x/3 - y/12 - 19/4 = 0`

Solve the system of equations by using the method of cross multiplication:

`1/x + 1/y = 7, 2/x + 3/y = 17`

Solve the system of equations by using the method of cross multiplication:

`5/(x + y) - 2/(x - y) + 1 = 0, 15/(x + y) + 7/(x - y) - 10 = 0`

Solve the system of equations by using the method of cross multiplication:

`(ax)/b - (by)/a - (a + b) = 0, ax - by - 2ab = 0`

Solve the system of equations by using the method of cross multiplication:

2ax + 3by – (a + 2b) = 0, 3ax + 2by – (2a + b) = 0

Solve the system of equations by using the method of cross multiplication:

`a/x - b/y = 0, (ab^2)/x + (a^2b)/y = (a^2 + b^2)`, where x ≠ 0 and y ≠ 0

Show that the following system of equations has a unique solution:

3x + 5y = 12, 5x + 3y = 4

Also, find the solution of the given system of equations.

Show that the following system of equations has a unique solution:

2x – 3y = 17, 4x + y = 13

Also, find the solution of the given system of equations.

Show that the following system of equations has a unique solution:

`x/3 + y/2 = 3, x - 2y = 2`

Also, find the solution of the given system of equations.

Find the value of k for which the following system of equations has a unique solution:

2x + 3y – 5 = 0, kx – 6y – 8 = 0

Find the value of k for which the following system of equations has a unique solution:

x – ky = 2, 3x + 2y + 5 = 0

Find the value of k for which the following system of equations has a unique solution:

5x – 7y – 5 = 0, 2x + ky – 1 = 0

Find the value of k for which the following system of equations has a unique solution:

4x + ky + 8 = 0, x + y + 1 = 0

Find the value of k for which the following system of equations has a unique solution:

4x – 5y = k, 2x – 3y = 12

Find the value of k for which the following system of equations has a unique solution:

kx + 3y = (k – 3), 12x + ky = k

Show that the system of equations

2x – 3y = 5, 6x – 9y = 15

has an infinite number of solutions.

Show that the system of equations

`6x + 5y = 11, 9x + 15/2 y = 21`

has no solution.

For what value of k does the system of equations

kx + 2y = 5, 3x – 4y = 10

have (i) a unique solution, (ii) no solution?

For what value of k does the system of equations

x + 2y = 5, 3x + ky + 15 = 0

have (i) a unique solution, (ii) no solution?

For what value of k does the system of equations

x + 2y = 3, 5x + ky + 7 = 0

have (i) a unique solution, (ii) no solution?

Also, show that there is no value of k for which the given system of equations has infinitely many solutions.

Find the value of k for which the following system of linear equations has an infinite number of solutions:

2x + 3y = 7, (k – 1)x + (k + 2)y = 3k

Find the value of k for which the following system of linear equations has an infinite number of solutions:

2x + (k – 2)y = k, 6x + (2k – 1)y = (2k + 5)

Find the value of k for which the following system of linear equations has an infinite number of solutions:

kx + 3y = (2k + 1), 2(k + 1)x + 9y = (7k + 1)

Find the value of k for which the following system of linear equations has an infinite number of solutions:

5x + 2y = 2k, 2(k + 1)x + ky = (3k + 4)

Find the value of k for which the following system of linear equations has an infinite number of solutions:

(k – 1)x – y = 5, (k + 1)x + (1 – k)y = (3k + 1)

Find the value of k for which the following system of linear equations has an infinite number of solutions:

(k – 3)x + 3y – k, kx + ky – 12 = 0

Find the values of a and b for which the following system of linear equations has an infinite number of solutions:

(a – 1)x + 3y = 2, 6x + (1 – 2b)y = 6

Find the values of a and b for which the following system of linear equations has an infinite number of solutions:

(2a – 1)x + 3y = 5, 3x + (b – 1)y = 15

Find the values of a and b for which the following system of linear equations has an infinite number of solutions:

2x – 3y = 7, (a + b)x – (a + b – 3)y = 4a + b

Find the values of a and b for which the following system of linear equations has an infinite number of solutions:

2x + 3y = 7, (a + b + 1)x – (a + 2b + 2)y = 4(a + b) + 1

Find the values of a and b for which the following system of linear equations has an infinite number of solutions:

2x + 3y = 7, (a + b)x + (2a – b)y = 21

Find the values of a and b for which the following system of linear equations has an infinite number of solutions:

2x + 3y = 7, 2ax + (a + b)y = 28

Find the value of k for which the system of equations

8x + 5y = 9, kx + 10y = 15

has a non-zero solution.

Find the value of k for which the following system of equations has no solution:

kx + 3y = 3, 12x + ky = 6

Find the value of k for which the following system of equations has no solution:

3x – y – 5 = 0, 6x – 2y + k = 0 (k ≠ 0)

Find the value of k for which the system of equations

kx + 3y + 3 – k = 0, 12x + ky – k = 0 has no solution.

Find the value of k for which the system of equations

5x – 3y = 0, 2x + ky = 0

has a non-zero solution.

5 chairs and 4 tables together cost ₹ 5600, while 4 chairs and 3 tables together cost ₹ 4340. Find the cost of a chair and that of a table.

23 spoons and 17 forks together cost Rs. 1770, while 17 spoons and 23 forks together cost Rs. 1830. Find the cost of a spoon and that of a fork.

A lady has only 25-paisa coins and 50-paisa coins in her purse. If she has 50 coins in all totalling Rs. 19.50, how many coins of each kind does she have?

The sum of two numbers is 137 and their difference is 43. Find the numbers.

Find two numbers such that the sum of twice the first and thrice the second is 92 and four times the first exceeds seven times the second by 2.

Find two numbers such that the sum of thrice the first and the second is 142 and four times the first exceeds the second by 138.

If 45 is subtracted from twice the greater of two numbers, it results in the other number. If 21 is subtracted from twice the smaller number, it results in the greater number. Find the numbers.

If three times the larger of two numbers is divided by the smaller, we get 4 as the quotient and 8 as the remainder. If five times the smaller is divided by the larger, we get 3 as the quotient and 5 as the remainder. Find the numbers.

If 2 is added to each of two given numbers, their ratio becomes 1 : 2. However, if 4 is subtracted from each of the given numbers, the ratio becomes 5 : 11. Find the numbers.

The difference between two numbers is 14 and the difference between their squares is 448. Find the numbers.

The sum of the digits of a two-digit number is 12. The number obtained by interchanging its digits exceeds the given number by 18. Find the number.

A number consisting of two digits is seven times the sum of its digits. When 27 is subtracted from the number, the digits are reversed. Find the number.

The sum of the digits of a two-digit number is 15. The number obtained by interchanging the digits exceeds the given number by 9. Find the number.

A two-digit number is 3 more than 4 times the sum of its digits. If 18 is added to the number, the digits are reversed. Find the number.

A number consists of two digits. When it is divided by the sum of its digits, the quotient is 6 with no remainder. When the number is diminished by 9, the digits are reversed. Find the number.

A two-digit number is such that the product of its digits is 35. If 18 is added to the number, the digits interchange their places. Find the number.

A two-digit number is such that the product of its digits is 18. When 63 is subtracted from the number, the digits interchange their places. Find the number.

The sum of a two-digit number and the number obtained by reversing the order of its digits is 121, and the two digits differ by 3. Find the number.

The sum of the numerator and denominator of a fraction is 8. If 3 is added to both of the numerator and the denominator, the fraction becomes `3/4`. Find the fraction.

If 2 is added to the numerator of a fraction, it reduces to `(1/2) `and if 1 is subtracted from the denominator, it reduces to `(1/3)`. Find the fraction.

The denominator of a fraction is greater than its numerator by 11. If 8 is added to both its numerator and denominator, it becomes `3/4`. Find the fraction.

Find a fraction which becomes `(1/2)` when 1 is subtracted from the numerator and 2 is added to the denominator, and the fraction becomes `(1/3)` when 7 is subtracted from the numerator and 2 is subtracted from the denominator.

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.

The sum of two numbers is 16 and the sum of their reciprocals is `1/3`. Find the numbers.

There are two classrooms A and B. If 10 students are sent from A to B, the number of students in each room becomes the same. If 20 students are sent from B to A, the number of students in A becomes double the number of students in B. Find the number of students in each room.

Taxi charges in a city consist of fixed charges per day and the remaining depending upon the distance travelled in kilometres. If a person travels 80 km, he pays Rs. 1330, and for travelling 90 km, he pays Rs. 1490. Find the fixed charges per day and the rate per km.

A part of monthly hostel charges in a college are fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for 25 days, he has to pay Rs. 4550 as hostel charges whereas a student B, who takes food for 30 days, pays Rs. 5200 as hostel charges. Find the fixed charges and the cost of the food per day.

A man invested an amount at 10% per annum simple interest and another amount at 8% per annum simple interest. He received an annual interest of Rs. 1350. But, if he had interchanged the amounts invested, he would have received Rs. 45 less. What amounts did he invest at different rates?

The monthly incomes of A and B are in the ratio of 5 : 4 and their monthly expenditures are in the ratio of 7 : 5. If each saves Rs. 9000 per month, find the monthly income of each.

A man sold a chair and a table together for Rs. 1520, thereby making a profit of 25% on 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.

Points A and B are 70 km apart 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 1 hour. Find the speed of each car.

A train covered a certain distance at a uniform speed. If the train had been 5 kmph faster, it would have taken 3 hours less than the scheduled time. And, if the train were slower by 4 kmph, it would have taken 3 hours more than the scheduled time. Find the length of the journey.

Abdul travelled 300 km by train and 200 km by taxi taking 5 hours and 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 taxi.

Places A and B are 160 km apart 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 8 hours. But, if they travel towards each other, they meet in 2 hours. Find the speed of each car.

A sailor goes 8 km downstream in 40 minutes and returns in 1 hour. Find the speed of the sailor in still water and the speed of the current.

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 downstream. Determine the speed of the stream and that of the boat in still water.

2 men and 5 boys can finish a piece of work in 4 days, while 3 men and 6 boys can finish it in 3 days. Find the time taken by one man alone to finish the work and that taken by one boy alone to finish the work.

The length of a room exceeds its breadth by 3 meters. If the length is increased by 3 meters and the breadth is decreased by 2 meters, the area remains the same. Find the length and the breadth of the room.

The area of a rectangle gets reduced by 8 m², when its length is reduced by 5 m and its breadth is increased by 3 m. If we increase the length by 3 m and breadth by 2 m, the area is increased by 74 m². Find the length and the breadth of the rectangle.

The area of a rectangle gets reduced by 67 square meters, when its length is increased by 3 m and the breadth is decreased by 4 m. If the length is reduced by 1 m and breadth is increased by 4 m, the area is increased by 89 square meters, Find the dimension of the rectangle.

A railway half ticket costs half the full fare and the reservation charge is the some on half ticket as on full ticket. One reserved first class ticket from Mumbai to Delhi costs ₹ 4150 while one full and one half reserved first class ticket cost ₹ 6255. What is the basic first class full fare and what is the reservation charge?

Five years hence, a man’s age will be three times the sum of the ages of his son. Five years ago, the man was seven times as old as his son. Find their present ages

Two years ago, a father was five times as old as his son. Two year later, his age will be 8 more than three times the age of the son. Find the present ages of father and son.

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.

On selling a tea set at 5% loss and a lemon set at 15% gain, a crockery seller gains ₹ 7. If he sells the tea set at 5% gain and the lemon set at 10% gain, he gains ₹ 13. Find the actual price of each of the tea set and the lemon set.

A lending library has fixed charge for the first three days and an additional charge for each day thereafter. Mona paid ₹ 27 for a book kept for 7 days, while Tanvy paid ₹ 21 for the book she kept for 5 days find the fixed charge and the charge for each extra day.

A number consists of two digits whose sum is 9. If 27 is added to the number, the digits change their places. Find the number.

A fraction becomes `(1)/(3)` when 2 is subtracted from the numerator and it becomes `(1)/(2)` when 1 is subtracted from the denominator. Find the fraction.

A father’s age is three times the sum of the ages of his two children. After 5 years his age will be two times the sum of their ages. Find the present age of the father.

The sum of the areas of two squares is 640 m². If the difference in their perimeter be 64 m, find the sides of the two squares.

Sum of the areas of two squares is 157 m². If the sum of their perimeters is 68 m, find the sides of the two squares.

A chemist has one solution containing 50% acid and a second one containing 25% acid. How much of each should be used to make 10 litres of a 40% acid solution?

A jeweller has bars of 18-carat gold and 12-carat gold. How much of each must be melted together to obtain a bar of 16-carat gold, weighing 120 gm? (Given: Pure gold is 24-carat).

90% and 97% pure acid solutions are mixed to obtain 21 litres of 95% pure acid solution. Find the quantity of each type of acid to be mixed to form the mixture.

The larger of the two supplementary angles exceeds the smaller by 180°. Find them.

In a ΔАBC, ∠A = x°, ∠B = (3x – 2)°, ∠C = y° and ∠C – ∠B = 9°. Find the three angles.

In a cyclic quadrilateral ABCD, it is given that ∠A = (2x + 4)°, ∠B = (y + 3)°, ∠C = (2y + 10)° and ∠D = (4x – 5)°. Find the four angles.

Write the number of solutions of the following pair of linear equations:

x + 2y – 8 = 0, 2x + 4y = 16

Find the value of k for which the following pair of linear equations have infinitely many solutions:

2x + 3y = 7, (k – 1)x + (k + 2)y = 3k

For what value of k does the following pair of linear equations have infinitely many solutions?

10x + 5y – (k – 5) = 0 and 20x + 10y – k = 0

For what value of k will the following pair of linear equations have no solution?

2x + 3y = 9, 6x + (k – 2)y = (3k – 2)

Write the number of solutions of the following pair of linear equations:

x + 3y – 4 = 0, 2x + 6y – 7 = 0

Write the value of k for which the system of equations 3x + ky = 0, 2x – y = 0 has a unique solution.

The difference between two numbers is 5 and the difference between their squares is 65. Find the numbers.

The cost of 5 pens and 8 pencils is ₹ 120, while the cost of 8 pens and 5 pencils is ₹ 153. Find the cost of 1 pen and that of 1 pencil.

The sum of two numbers is 80. The larger number exceeds four times the smaller one by 5. Find the numbers.

A number consists of two digits whose sum is 10. If 18 is subtracted form the number, its digits are reversed. Find the number.

A man purchased 47 stamps of 20 p and 25 p for ₹ 10. Find the number of each type of stamps.

A man has some hens and cows. If the number of heads be 48 and number of feet by 140, how many cows are there?

If `2/x + 3/y = 9/(xy)` and `4/x +  9/y = 21/(xy)`, find the values of x and y.

If `x/4 + y/3 = 5/12` and `x/2 + y = 1`, then find the value of (x + y).

If 12x + 17y = 53 and 17x + 12y = 63 then find the value of (x + y).

Find the value of k for which the system of equations 3x + 5y = 0 and kx + 10y = 0 has infinite nonzero solutions.

Find the value of k for which the system of equations kx – y = 2 and 6x – 2y = 3 has a unique solution.

Find the value of k for which the system of equations 2x + 3y – 5 = 0 and 4x + ky – 10 = 0 has infinite number of solutions.

Show that the system 2x + 3y – 1= 0 and 4x + 6y – 4 = 0 has no solution.

Find the value of k for which the system of equations x + 2y – 3 = 0 and 5x + ky + 7 = 0 is inconsistent.

Solve the following system of equations:

`3/(x + y) + 2/(x - y) = 2`

`9/(x + y) - 4/(x - y) = 1`

If 2x + 3y = 12 and 3x – 2y = 5 then

If x – y = 2 and `2/(x + y) = 1/5` then

If `(2x)/3 - y/2 + 1/6 = 0` and `x/2 + (2y)/3 = 3` then

If `1/x + 2/y = 4` and `3/y - 1/x = 11` then

If `(2x + y + 2)/5 = (3x - y + 1)/3 = (3x + 2y + 1)/6` then

If `3/(x + y) + 2/(x - y) = 2` and `9/(x + y) - 4/(x - y) = 1` then

If 4x + 6y = 3xy and 8x + 9y = 5xy then

If 29x + 37y = 103 and 37x + 29y = 95 then

If `2^(x + y) = 2^(x - y) = sqrt(8)` then the value of y is ______.

If `2/x + 3/y = 6` and `1/x + 1/(2y) = 2` then

The system kx – y = 2 and 6x – 2y = 3 has a unique solution only when

The system x – 2y = 3 and 3x + ky = 1 has a unique solution only when

The system x + 2y = 3 and 5x + ky + 7 = 0 has no solution, when

If the lines given by 3x + 2ky = 2 and 2x + 5y + 1 = 0 are parallel then the value of k is ______.

For what value of k do the equations kx – 2y = 3 and 3x + y = 5 represent two lines intersecting at a unique point?

The pair of equations x + 2y + 5 = 0 and –3x – 6y + 1 = 0 has

The pair of equations 2x + 3y = 5 and 4x + 6y = 15 has

If a pair of linear equations is consistent then their graph lines will be

If a pair of linear equations is inconsistent then their graph lines will be

In a ΔАВС, ∠C = 3 ∠B = 2(∠A + ∠B), then ∠B = ?

In a cyclic quadrilateral ABCD, it is being given that ∠A = (x + y + 10)°, ∠B = (y + 20)°, ∠C = (x + y – 30)° and ∠D = (x + y)°. Then, ∠B = ?

The sum of the digits of a two-digit number is 15. The number obtained by interchanging the digits exceeds the given number by 9. The number is ______.

In a given fraction, if 1 is subtracted from the numerator and 2 is added to the denominator, it becomes `1/2`. If 7 is subtracted from the numerator and 2 is subtracted from the denominator, it becomes `1/3`. The fraction is ______.

5 years hence, the age of a man shall be 3 times the age of his son while 5 years earlier the age of the man was 7 times the age of his son. The present age of the man is ______.

The graphs of the equations 6x – 2y + 9 = 0 and 3x – y + 12 = 0 are two lines which are

The graphs of the equations 2x + 3y – 2 = 0 and x – 2y – 8 = 0 are two lines which are

The graphs of the equations 5x – 15y = 8 and 3x – 9y = `24/5` are two lines which are

The graphic representation of the equations x + 2y = 3 and 2x + 4y + 7 = 0 gives a pair of

If 2x – 3y = 7 and (a + b)x – (a + b – 3)y = 4a + b have an infinite number of solutions then

The pair of equations 2x + y = 5, 3x + 2y = 8 has

Show that the system of equations –x + 2y + 2 = 0 and `1/2x - 1/2y - 1 = 0` has a unique solution.

For what values of k is the system of equations kx + 3y = k – 2, 12x + ky = k inconsistent?

Show that the equations `9x - 10y = 21, (3x)/2 - (5y)/3 = 7/2` have infinitely many solutions.

Solve the system of equations x – 2y = 0, 3x + 4y = 20.

Show that the paths represented by the equations x – 3y = 2 and –2x + 6y = 5 are parallel.

The difference between two numbers is 26 and one number is three times the other. Find the numbers.

Solve: 23x + 29y = 98, 29x + 23y = 110.

Solve: 6x + 3y = 7xy and 3x + 9y = 11xy.

Find the value of k for which the system of equations 3x + y = 1 and kx + 2y = 5 has (i) a unique solution, (ii) no solution.

In а ΔАВС, ∠C = 3∠B = 2(∠A + ∠B). Find the measure of each one of ∠A, ∠B and C.

5 pencils and 7 pens together cost ₹ 195 while 7 pencils and 5 pens together cost ₹ 153. Find the cost of each one of the pencil and the pen.

Solve the following system of equations graphically:

2x – 3y = 1, 4x – 3y + 1 = 0

Find the angles of a cyclic quadrilateral ABCD in which ∠A = (4x + 20)°, ∠B = (3x – 5)°, ∠C = (4y)° and ∠D = (7y + 5)°.

Solve for x and y:

`35/(x + y) + 14/(x - y) = 19, 14/(x + y) + 35/(x - y) = 37`

If 1 is added to both the numerator and the denominator of a fraction, it becomes `4/5`. If, however, 5 is subtracted from both the numerator and the denominator, the fraction becomes `1/2`. Find the fraction.

Solve: `(ax)/b - (by)/a = a + b, ax - by = 2ab`.