Pair Of Linear Equations In Two Variables Class 10th Mathematics Gujarat Board Solution

Class 10th Mathematics Gujarat Board Solution
Exercise 3.1
  1. Father tells his son, "Five years ago, I was seven times as old as you were.…
  2. The sum of the cost of 1kg apple and 1kg pine-apple is Rs. 150 and the cost of 1…
  3. Nilesh got twice the marks as obtained by Ilesh, in the annual examination of…
  4. Length of a rectangle is four less than the thrice of its breadth. The perimeter…
  5. The sum of the weights of a father and a son is 85 kg. The weight of the son is…
  6. In a cricket match, Sachin Tendulkar makes his score thrice the Sehwag's score.…
  7. In tossing a balanced coin, the probability of getting head on its face is twice…
Exercise 3.2
  1. 2x + y = 8, x + 6y = 15 Solve the following pair of linear equation in two…
  2. x + y = 1, 3x + 3y = 2 Solve the following pair of linear equation in two…
  3. 2x + 3y = 5, x + y = 2 Solve the following pair of linear equation in two…
  4. x-y = 6, 3x - 3y = 18 Solve the following pair of linear equation in two…
  5. (x + 2) (y — 1) = xy, (x — 1) (y + 1) = xy Solve the following pair of linear…
  6. Draw the graphs of lie pair of linear equations 3x + 2y = 5 and 2x-3y = -1.…
  7. 15 students of class X took part in the examination of Indian mathematics…
  8. Examine graphically whether the pair of equations 2x + 3y = 5 and x + 9/6 y =…
Exercise 3.3
  1. x + y = 7, 3x — y = 1 Solve the following pairs of linear equations by the…
  2. 3x — y = 0, x — y + 6 = 0 Solve the following pairs of linear equations by the…
  3. 2x + 3y = 5, 2x + 3y = 7 Solve the following pairs of linear equations by the…
  4. x — y = 3, 3x — 3y = 9 Solve the following pairs of linear equations by the…
  5. 3x/2 - 5y/3 = - 2 , x/3 + y/2 = 13/6 Solve the following pairs of linear…
  6. Solve the pair of linear equations x — y = 28 and x — 3y = 0 and if the solution…
  7. A fraction becomes 4/5 if 3 is added to both the numerator and the denominator.…
  8. The sum of present ages of a father and his son is 50 years. After 5 years, the…
  9. A bus traveller travelling with some of his relatives buys 5 tickets from…
Exercise 3.4
  1. x/5 - y/3 = 4/15 , x/2 - y/9 = 7/18 Solve the following pair of linear…
  2. 4x — 19y + 13 = 0, 13x — 23y = —19 Solve the following pair of linear equations…
  3. x + y = a + b, ax — by = a^2 — b^2 Solve the following pair of linear equations…
  4. 5ax + 6by = 28; 3ax + 4by = 18 Solve the following pair of linear equations by…
  5. The sum of two numbers is 35. Four times the larger number is 5 more than 5…
  6. There are some 25 paise coins and some 50 paise coins in a bag. The total number…
  7. The sum of the digits of two-digit number is 3. The number obtained by…
  8. The length of a rectangle is twice its breadth. The perimeter of the rectangle…
  9. An employee deposits certain amount at the rate of 8% per annum and a certain…
Exercise 3.5
  1. 0.3x + 0.4y = 2.5 and 0.5x — 0.3y = 0.3 Solve the following pairs of equations…
  2. 5x + 8y = 18, 2x — 3y = 1 Solve the following pairs of equations by cross…
  3. x/3 + y/5 = 1 , 7x— 15y = 21 Solve the following pairs of equations by cross…
  4. 3x + y = 5, 5x + 3y = 3 Solve the following pairs of equations by cross…
  5. By cross multiplication method, find such a two digit number such that, the…
  6. The sum of two numbers is 70 and their difference is 6. Find these numbers by…
  7. While arranging certain students of a school in rows containing equal number of…
  8. In AABC, the measure of ∠B is thrice to the measure of ∠C and the measure of ∠A…
Exercise 3.6
  1. 5/2x + 2/3y = 7 , 3/x + 2/y = 12 x ≠ 0, y ≠ 0 Solve the following pairs of…
  2. 2x + 3y = 2xy, 6x + 12y = 7xy Solve the following pairs of linear equations:…
  3. 4/x-1 + 5/y-1 = 2 , 8/x-1 + 15/y-1 = 3 x ≠ 1, y ≠ 1 Solve the following pairs…
  4. 1/3x+y + 1/3x-y = 3/4 , 1/2 (3x+y) - 1/2 (3x-y) = -1/8 3x + y ≠ 0, 3x - y ≠ 0…
  5. 3/root x + 4/root y = 2 , 5/root x + 7/root y = 41/42 x 0, y 0 Solve the…
  6. 5 women and 2 men together can finish an embroidery work in 4 days, while 6…
  7. A boat goes 21 km upstream and 18 km downstream in 9 hours. In 13 hours, it can…
  8. Solve the following pair of equations by cross multiplication method: 4x+7y/xy =…
  9. Mahesh travels 250 km to his home partly by train and partly by bus. He takes 6…
Exercise 3
  1. Obtain a pair of linear equations from the following information: "The rate of…
  2. Draw the graphs of the pair of linear equations in two variables. x + 3y = 6, 2x…
  3. Solve the following pair of equations by the method of elimination: 4/x + 5/y =…
  4. Solve the following pair of linear equations by the method of cross -…
  5. Solve the following pair of equations: 4/x+1 + 7/y+2 = 2 and 10/x+1 + 14/y+2 =…
  6. The difference between two natural numbers is 6. Adding 10 to the twice of the…
  7. The area of a rectangle gets increased by 30 square units, if its length is…
  8. A part of monthly hostel charges is fixed and the remaining depends on the…
  9. A fraction becomes 2/5 when 2 is subtracted from the numerator and denominator…
  10. The solution set of x — 3y = 1 and 3x + y = 3 isA. {(0, 1)) B. {(1, 1)} C.…
  11. The solution set of 2x + y = 6 and 4x + 2y = 5 isA. {(x, Y)| 2x + y = 6; x, y…
  12. To eliminate x, from 3x + y = 7 and —x + 2y = 2 second equation is multiplied…
  13. If 2x + 3y = 7 and 3x + 2y = 3, then x — y =A. 4 B. —4 C. 2 D. —2…
  14. If the pair of linear equations ax + 2y = 7 and 2x + 3y = 8 has a unique…
  15. The pair of linear equations 2x + y — 3 = 0 and 6x + 3y = 9 hasA. a unique…
  16. If in a two digit number, the digit at unit place is x and the digit at tens…
  17. In a two digit number, the digit at tens place is 7 and the sum of the digits…
  18. The sum of two numbers is 10 and the difference of them is 2. Then the greater…
  19. 3 years ago, the sum of ages of a father and his son was 40 years. After 2…
  20. The solution set of 2x + 4y = 8 and x + 2y = 4 isA. {(2, 1)} B. empty set C.…
  21. Equation x/2 - y/3 = 1 can be expressed in the standard form asA. 2x — 3y — 6…

Exercise 3.1
Question 1.

Obtain a pair of linear equations in two variables from the following information:

Father tells his son, "Five years ago, I was seven times as old as you were. After five years, I will be three times as old as you will be".


Answer:

Let the present age of father = x


Let the present age of son = y


Now,


Given, Five years ago, father was seven times as old as son.


⇒ (x–5) = 7(y–5)


x – 5 = 7y – 35


= x – 7y + 30 = 0 …[1]


Also, after five years, father will be three times as old as son


⇒ (x + 5) = 3(y + 5)


x + 5 = 3y + 15


= x – 3y – 10 = 0 …[2]


eq[1] and eq[2] are required equations



Question 2.

Obtain a pair of linear equations in two variables from the following information:

The sum of the cost of 1kg apple and 1kg pine-apple is Rs. 150 and the cost of 1 kg apple is twice the cost of 1kg pine-apple.


Answer:

Let the cost of 1 kg apple = x


Let the cost of 1 kg pine-apple = y


Given that, sum of the cost of 1 kg apple and 1 kg pine-apple is Rs. 150


⇒ x + y = 150 …[1]


Also, cost of 1 kg apple is twice the cost of 1kg pine-apple


⇒ x = 2y …[2]


eq[1] and eq[2] are the required equations.



Question 3.

Obtain a pair of linear equations in two variables from the following information:

Nilesh got twice the marks as obtained by Ilesh, in the annual examination of mathematics of standard 10. The sum of the marks as obtained by them is 135.


Answer:

let marks obtained by Ilesh be x and marks obtained by Nilesh by y.

Now,


given, Nilesh got twice the marks as obtained by Ilesh.


⇒ y = 2x …[1]


Also, the sum of the marks as obtained by them is 135.


x + y = 135 …[2]


These are the two required linear equations.



Question 4.

Length of a rectangle is four less than the thrice of its breadth. The perimeter of the rectangle is 110.


Answer:

Let the length of the rectangle be x and breadth be y.

Now,


given, Length of a rectangle is four less than the thrice of its breadth


x = 3y – 4 …[1]


and


also, the perimeter is 110


2(x + y) = 110


⇒ x + y = 55 …[2]


eq[1] and eq[2] are the required equations.



Question 5.

Obtain a pair of linear equations in two variables from the following information:

The sum of the weights of a father and a son is 85 kg. The weight of the son is 1/4 of the weight of his father.


Answer:

let x and y be the weight of father and son respectively.

Now,


Given, The sum of the weights of a father and a son is 85 kg


x + y = 85 …[1]


and,


also, The weight of the son is  of the weight of his father


 …[2]


These are the required linear equations.



Question 6.

Obtain a pair of linear equations in two variables from the following information:

In a cricket match, Sachin Tendulkar makes his score thrice the Sehwag's score. Both of them together make a total score of 200 runs.


Answer:

Let the score of Sachin and Sehwag be x and y respectively

given, Both of them together make a total score of 200 runs.


⇒ x + y = 200 …[1]


also, Sachin Tendulkar makes his score thrice the Sehwag's score


⇒ x = 3y …[2]


These are the two required equations.



Question 7.

Obtain a pair of linear equations in two variables from the following information:

In tossing a balanced coin, the probability of getting head on its face is twice to the probability of getting tail on its face. The sum of both probabilities (head and tail) is 1.


Answer:

Let the probabilities of getting heads and tails are x and y respectively.

Given, the probability of getting head on its face is twice to the probability of getting tail on its face


⇒ x = 2y …[1]


also, the sum of probabilities is 1


⇒ x + y = 1 …[2]


∴ these two equations are the required equations.




Exercise 3.2
Question 1.

Solve the following pair of linear equation in two variables (by graph)

2x + y = 8, x + 6y = 15


Answer:

Generally, we substitute x = 0 or y = 0 in the given linear equations to get y and x. So we get two points on the straight line. To find more points on the line, take different values of x related to it, we get different values for y from the equation.


We get the following tables for the given linear equations.


For 2x + y = 8


y = 8 – 2x



For x + 6y = 15


x = 15 – 6y




Plotting these points on a graph and joining them, we get two straight lines.


From the graph, we can see that both lines intersect at (3, 2), hence the solution to this pair is (3, 2).



Question 2.

Solve the following pair of linear equation in two variables (by graph)

x + y = 1, 3x + 3y = 2


Answer:

Generally, we substitute x = 0 or y = 0 in the given linear equations to get y and x. So we get two points on the straight line. To find more points on the line, take different values of x related to it, we get different values for y from the equation.

for x + y = 1


⇒ y = 1 – x



for, 3x + 3y = 2



Plotting these points on a graph and joining them, we get two straight lines.



From the graph, we can see that both lines are parallel and have no intersection points.



Question 3.

Solve the following pair of linear equation in two variables (by graph)

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


Answer:

Generally, we substitute x = 0 or y = 0 in the given linear equations to get y and x. So we get two points on the straight line. To find more points on line, take different values of x related to it, we get different values for y from the equation.

We get the following tables for the given linear equations.


for


2x + 3y = 5



for


x + y = 2



Plot it



Plotting these points on a graph and joining them, we get two straight lines.


From the graph, we can see that both lines intersect at (1, 1), hence solution to this pair is (1, 1).



Question 4.

Solve the following pair of linear equation in two variables (by graph)

x–y = 6, 3x – 3y = 18


Answer:

Generally, we substitute x = 0 or y = 0 in the given linear equations to get y and x. So we get two points on the straight line. To find more points on line, take different values of x related to it, we get different values for y from the equation.

We get the following tables for the given linear equations.


for


x—y = 6



for, 3x—3y = 18



Plotting these points on a graph and joining them, we get two straight lines.



From the graph, we can see that both lines coincide and hence there are infinite number of solutions.



Question 5.

Solve the following pair of linear equation in two variables (by graph)

(x + 2) (y — 1) = xy, (x — 1) (y + 1) = xy


Answer:

Generally, we substitute x = 0 or y = 0 in the given linear equations to get y and x. So we get two points on the straight line. To find more points on the line, take different values of x related to it, we get different values for y from the equation.

We get the following tables for the given linear equations.


for


(x + 2) (y — 1) = xy


It is a straight line



for,


(x — 1) (y + 1) = xy




Plotting these points on a graph and joining them, we get two straight lines.


From the graph, we can see that both lines intersect at (4, 3), hence the solution to this pair is (4, 3).



Question 6.

Draw the graphs of lie pair of linear equations 3x + 2y = 5 and 2x–3y = –1. Determine the coordinates of the vertices of the triangle formed by these linear equations and the X-axis.


Answer:

Generally, we substitute x = 0 or y = 0 in the given linear equations to get y and x. So, we get two points on the straight line. To find more points on line, take different values of x related to it, we get different values for y from the equation.

We get the following tables for the given linear equations.


for


3x + 2y = 5



for, 2x–3y = –1




when y = 0, lines would intersect the x-axis


so the coordinates of triangle at x-axis would be 


So the coordinates of the triangle will be




Question 7.

15 students of class X took part in the examination of Indian mathematics Olympiad. The number of boys participants is 5 less than the number of girls participants. Find the number of boys and girls (using a graph) who took part in the examination of Indian mathematics Olympiad.


Answer:

Let the number of girls be x and the number of boys be y.

Given, total no. of participants is 15.


⇒ x + y = 15


Also, boys participants is 5 less than the number of girls participants


⇒ x = y – 5


We have two equations we will plot them and their intersection will give the required result.


x + y = 15



x = y – 5




Plotting these points on a graph and joining them, we get two straight lines.


From the graph, we can see that both lines intersect at (5, 10), hence solution to this pair is (5, 10).


∴ no. of boys is 5 and no. of girls is 10



Question 8.

Examine graphically whether the pair of equations 2x + 3y = 5 and  is consistent


Answer:

Generally, we substitute x = 0 or y = 0 in the given linear equations to get y and x. So we get two points on the straight line. To find more points on line, take different values of x related to it, we get different values for y from the equation.

We get the following tables for the given linear equations.


for


2x + 3y = 5



for 



Plotting these points on a graph and joining them, we get two straight lines.