Welcome to the NCERT Solutions for Class 10th Mathematics - Chapter Pair of Linear Equations in Two Variables. This page offers a step-by-step solution to the specific question from Exercise 3, Question 1: **solve the following pair of linear equations by th...**.

Question 1

Solve the following pair of linear equations by the substitution method.

Answer

(i) x + y = 14 …………….(1)

x – y = 4 …………….(2)

From the equation (1), we get

x = 14 - y …………….(3)

Putting the value of x in equation (2), we get

(14 - y) – y = 4

14 – y – y = 4

- 2y = - 10

Putting the value of y in equation (3),

x = 14 – 5

x = 9

Hence, x = 9 and y = 5

(iii) 3x - y = 3 …………….(1)

9x – 3y = 9…………….(2)

From the equation (1), we get

Putting the value of y in equation (2), we get

9x – 3 (3x - 3) = 9

9x – 9x + 9 = 9

9 = 9, which is true.

Therefore, pair of linear equation has infinite many solutions.

- Q:-
Given the linear equation 2x + 3y – 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is:

(i) intersecting lines (ii) parallel lines (iii) coincident lines - Q:-
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. - Q:-
Solve 2x + 3y = 11 and 2x – 4y = – 24 and hence find the value of ‘m’ for which y = mx + 3.

- Q:-
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.

- Q:-
Form the pair of linear equations for the following problems and find their solution by substitution method.

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

(ii) The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.

(iii) The coach of a cricket team buys 7 bats and 6 balls for Rs. 3800. Later, she buys 3 bats and 5 balls for Rs. 1750. Find the cost of each bat and each ball.(iv) The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge paid is Rs. 105 and for a journey of 15 km, the charge paid is Rs. 155. What are the fixed charges and the charge per km? How much does a person have to pay for travelling a distance of 25 km?

(v) A fraction becomes, , if 2 is added to both the numerator and the denominator. If, 3 is added to both the numerator and the denominator it becomes Find the fraction.(vi) 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?

- Q:-
The coach of a cricket team buys 3 bats and 6 balls for ` 3900. Later, she buys another bat and 3 more balls of the same kind for ` 1300. Represent this situation algebraically and geometrically.

- Q:-
The cost of 2 kg of apples and 1kg of grapes on a day was found to be ` 160. After a month, the cost of 4 kg of apples and 2 kg of grapes is ` 300. Represent the situation algebraically and geometrically.

- Q:-
Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be.” (Isn’t this interesting?) Represent this situation algebraically and graphically.

- Q:-
Use Euclid’s division algorithm to find the HCF of :

(i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255 - Q:-
The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

- Q:-
A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 30° (see Fig. 9.11).

- Q:-
Complete the following statements:

(i) Probability of an event E + Probability of the event ‘not E’ =

(ii) The probability of an event that cannot happen is

(iii) The probability of an event that is certain to happen is

(iv) The sum of the probabilities of all the elementary events of an experiment is(v) The probability of an event is greater than or equal to

- Q:-
Check whether the following are quadratic equations :

(i) (x + 1)^{2}= 2(x – 3) (ii) x^{2}– 2x = (–2) (3 – x) (iii) (x – 2)(x + 1) = (x – 1)(x + 3) (iv) (x – 3)(2x +1) = x(x + 5)(v) (2x – 1)(x – 3) = (x + 5)(x – 1) (vi) x

^{2}+ 3x + 1 = (x – 2)^{2}(vii) (x + 2)^{3}= 2x (x2 – 1) (viii) x^{3}– 4x^{2}– x + 1 = (x – 2)^{3} - Q:-
How many tangents can a circle have?

- Q:-
Show that any positive odd integer is of the form 6

*q*+ 1, or 6*q*+ 3, or 6*q*+ 5, where*q*is some integer. - Q:-
A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.

- Q:-
Which of the following experiments have equally likely outcomes? Explain.

(i) A driver attempts to start a car. The car starts or does not start.

(ii) A player attempts to shoot a basketball. She/he shoots or misses the shot.

(iii) A trial is made to answer a true-false question. The answer is right or wrong.

(iv) A baby is born. It is a boy or a girl. - Q:-
Represent the following situations in the form of quadratic equations :

(i) The area of a rectangular plot is 528 m^{2}. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.(ii) The product of two consecutive positive integers is 306. We need to find the integers.

(iii) Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age.

(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.

- Q:-
A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball is double that of a red ball, determine the number of blue balls in the bag.

- Q:-
Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:

(i) t^{2}– 3, 2t^{4}+ 3t^{3}– 2t^{2}– 9t – 12(ii) x

^{2}+ 3x + 1, 3x^{4}+ 5x^{3}– 7x^{2}+ 2x + 2(iii) x

^{3}– 3x + 1, x^{5}– 4x^{3}+ x^{2}+ 3x + 1 - Q:-
Prove that the parallelogram circumscribing a circle is a rhombus.

- Q:-
Find the roots of the following equations:

- Q:-
Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:

(i) 2x

^{2 }– 3x + 5 = 0 (iii) 2x^{2}– 6x + 3 = 0 - Q:-
A game of chance consists of spinning an arrow which comes to rest pointing at one of the numbers 1, 2, 3, 4, 5, 6, 7, 8 (see Fig. 15.5 ), and these are equally likely outcomes. What is the probability that it will point at

(i) 8 ?

(ii) an odd number?

(iii) a number greater than 2?

(iv) a number less than 9? - Q:-
The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.

- Q:-
Fill in the blanks :

(i) A tangent to a circle intersects it in

(ii) A line intersecting a circle in two points is called a

(iii) A circle can have

(iv) The common point of a tangent to a circle and the circle is called - Q:-
Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m

^{2}? If so, find its length and breadth. - Q:-
Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, –7, –14 respectively.

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