NCERT Exemplar Solutions
Class 10 - Mathematics

CHAPTER 3: Pair of Linear Equations in Two Variables

Do the following pair of linear equations have no solution? Justify your answer.

(i) \(2x + 4y = 3\), \(12y + 6x = 6\)

(ii) \(x = 2y\), \(y = 2x\)

(iii) \(3x + y - 3 = 0\), \(\dfrac{2}{3}x + \dfrac{1}{2}y = 2\)

Do the following equations represent a pair of coincident lines? Justify your answer.

(i) \(3x + \dfrac{1}{7}y = 3\), \(7x + 3y = 7\)

(ii) \(-2x - 3y = 1\), \(6y + 4x = -2\)

(iii) \(\dfrac{x}{2} + \dfrac{y}{5} + \dfrac{5}{16} = 0\), \(4x + 8y + \dfrac{5}{4} = 0\)

Are the following pair of linear equations consistent? Justify your answer.

(i) \(-3x - 4y = 12\), \(4y + 3x = 12\)

(ii) \(\dfrac{3}{5}x - y = 12\), \(\dfrac{1}{5}x - 3y = 16\)

(iii) \(2ax + by = a\), \(4ax + 2by - 2a = 0\); \(a,b \ne 0\)

(iv) \(x + 3y = 11\), \(2(2x + 6y) = 22\)

For the pair of equations \(\lambda x + 3y = -7\) and \(2x + 6y = 14\) to have infinitely many solutions, the value of \(\lambda\) should be 1. Is the statement true? Give reasons.

For all real values of \(c\), the pair of equations \(x - 2y = 8\) and \(5x - 10y = c\) have a unique solution. Justify whether it is true or false.

The line represented by \(x = 7\) is parallel to the x–axis. Justify whether the statement is true or not.

Sample Question 1: Is it true to say that the pair of equations \(-x + 2y + 2 = 0\) and \(\dfrac{1}{2}x - \dfrac{1}{4}y = -1\) has a unique solution? Justify your answer.

Sample Question 2: Do the equations \(4x + 3y - 1 = 5\) and \(12x + 9y = 15\) represent a pair of coincident lines? Justify your answer.

Sample Question 3: Is the pair of equations \(x + 2y - 3 = 0\) and \(3x + 6y - 9 = 0\) consistent? Justify your answer.

For which value(s) of \(\lambda\) do the pair of linear equations \(\lambda x + y = \lambda^2\) and \(x + \lambda y = 1\) have (i) no solution, (ii) infinitely many solutions, (iii) a unique solution?

For which value(s) of \(k\) will the pair of equations \(kx + 3y = k - 3\) and \(12x + ky = k\) have no solution?

For which values of \(a\) and \(b\) will the pair of equations \(x + 2y = 1\) and \((a-b)x + (a+b)y = a + b - 2\) have infinitely many solutions?

Find the value(s) of p in (i) to (iv) and p and q in (v) for the following pair of equations:

(i) \(3x – y – 5 = 0\) and \(6x – 2y – p = 0\),
if the lines represented by these equations are parallel.

(ii) \(–x + py = 1\) and \(px – y = 1\),
if the pair of equations has no solution.

(iii) \(– 3x + 5y = 7\) and \(2px – 3y = 1\),
if the lines represented by these equations are intersecting at a unique point.

(iv) \(2x + 3y – 5 = 0\) and \(px – 6y – 8 = 0\), if the pair of equations has a unique solution.

(v) \(2x + 3y = 7\) and \(2px + py = 28 – qy\), if the pair of equations have infinitely many solutions.

The paths \(x - 3y = 2\) and \(-2x + 6y = 5\) represent straight lines. Do the paths cross each other?

Write a pair of linear equations whose unique solution is \(x = -1\), \(y = 3\). How many such pairs can you write?

If \(2x + y = 23\) and \(4x - y = 19\), find the values of \(5y - 2x\) and \(\dfrac{y}{x} - 2\).

In the rectangle, opposite sides are equal. Given the labels in Fig. 3.2, find \(x\) and \(y\): top \(= x + 3y\), bottom \(= 13\), left \(= 3x + y\), right \(= 7\).

(i). \(x + y = 3.3\) and \(\dfrac{0.6}{3x - 2y} = -1\), \(3x - 2y \ne 0\).

(ii). Solve: \(\dfrac{x}{3} + \dfrac{y}{4} = 4\) and \(\dfrac{5x}{6} - \dfrac{y}{8} = 4\).

(iii). Solve: \(4x + \dfrac{6}{y} = 15\) and \(6x - \dfrac{8}{y} = 14\), \(y \ne 0\).

(iv). Solve: \(\dfrac{1}{2x} - \dfrac{1}{y} = -1\) and \(\dfrac{1}{x} + \dfrac{1}{2y} = 8\), \(x,y \ne 0\).

(v). Solve: \(43x + 67y = -24\) and \(67x + 43y = 24\).

(vi). Solve: \(\dfrac{x}{a} + \dfrac{y}{b} = a + b\) and \(\dfrac{x}{a^2} + \dfrac{y}{b^2} = 2\), \(a,b \ne 0\).

(vii). Solve: \(\dfrac{2xy}{x + y} = \dfrac{3}{2}\) and \(\dfrac{xy}{2x - y} = -\dfrac{3}{10}\), with \(x + y \ne 0\) and \(2x - y \ne 0\).

Solve the pair \(\dfrac{x}{10} + \dfrac{y}{5} - 1 = 0\) and \(\dfrac{x}{8} + \dfrac{y}{6} = 15\). Hence, if \(y = \lambda x + 5\), find \(\lambda\).

By the graphical method, decide consistency and solve:
(i). \(3x + y + 4 = 0\) and \(6x - 2y + 4 = 0\) (ii). \(x - 2y = 6\) and \(3x - 6y = 0\) (iii). \(x + y = 3\) and \(3x + 3y = 9\)

Draw the graphs of \(2x + y = 4\) and \(2x - y = 4\). Find the vertices of the triangle formed by these two lines and the y-axis, and its area.

Find an equation of a line passing through the point that is the solution of \(x + y = 2\) and \(2x - y = 1\). How many such lines are there?

If \(x + 1\) is a factor of \(2x^3 + ax^2 + 2bx + 1\) and \(2a - 3b = 4\), find \(a\) and \(b\).

The angles of a triangle are \(x\), \(y\), and \(40^\circ\). Their difference \(|x - y|\) is \(30^\circ\). Find \(x\) and \(y\).

Two years ago, Salim was thrice his daughter's age. Six years later, he will be four years older than twice her age. Find their present ages.

A father's present age is twice the sum of the ages of his two children. After 20 years, his age will equal the sum of their ages then. Find the father's present age.

Two numbers are in the ratio \(5:6\). If 8 is subtracted from each, the ratio becomes \(4:5\). Find the numbers.

Students in halls A and B: if 10 go from A to B, they become equal. If 20 go from B to A, A becomes double B. Find the original numbers.

A shopkeeper charges a fixed amount for the first two days and a daily charge thereafter. Latika paid Rs 22 for 6 days; Anand paid Rs 16 for 4 days. Find the fixed charge and the daily charge after two days.

In a test, +1 mark for a correct answer and \(\dfrac{1}{2}\) mark deducted for a wrong answer. Jayanti answered 120 questions and scored 90 marks. How many did she answer correctly?

In cyclic quadrilateral \(ABCD\), \(\angle A = (6x + 10)^\circ\), \(\angle B = (5x)^\circ\), \(\angle C = (x + y)^\circ\), \(\angle D = (3y - 10)^\circ\). Find \(x\) and \(y\), then all four angles.

Graphically, solve the pair: \(2x + y = 6\) and \(2x - y + 2 = 0\). Also find the ratio of the areas of the two triangles formed by the pair of lines with (a) the x–axis and (b) the y–axis.

Determine, graphically, the vertices of the triangle formed by the lines \(y = x\), \(3y = x\), and \(x + y = 8\).

Draw the graphs of \(x = 3\), \(x = 5\), and \(2x - y - 4 = 0\). Find the area of the quadrilateral formed by these lines and the x–axis.

The cost of 4 pens and 4 pencil boxes is Rs 100. Also, three times the cost of a pen is Rs 15 more than the cost of a pencil box. Form the pair of linear equations and find both costs.

Determine, algebraically, the vertices of the triangle formed by the lines \(3x - y = 3\), \(2x - 3y = 2\), and \(x + 2y = 8\).

Ankita travels 14 km partly by rickshaw and partly by bus. She takes 30 minutes if 2 km is by rickshaw and the rest by bus. If 4 km is by rickshaw and the rest by bus, she takes 9 minutes longer. Find the speeds of the rickshaw and the bus.

A person rows at 5 km/h in still water. It takes thrice as much time to go 40 km upstream as 40 km downstream. Find the speed of the stream.

A motor boat covers 30 km upstream and 28 km downstream in 7 hours. It can travel 21 km upstream and return in 5 hours. Find its speed in still water and the speed of the stream.

A two-digit number equals \(8\) times the sum of its digits minus \(5\), and also equals \(16\) times the difference of its digits plus \(3\). Find the number.

A reserved first-class full ticket from A to B costs Rs 2530. A reserved full + a reserved half together cost Rs 3810. The reservation charge is the same for both, but a half ticket has half fare. Find the full fare and the reservation charge.

A shopkeeper sells a saree at 8% profit and a sweater at 10% discount to get Rs 1008 in total. If instead she sells the saree at 10% profit and the sweater at 8% discount, she gets Rs 1028. Find the cost price of the saree and the list price of the sweater.

Susan invests in two schemes A (8% p.a.) and B (9% p.a.). She receives Rs 1860 interest in total. If interchanged, the interest would be Rs 20 more. Find the amounts invested in each scheme.

Vijay sold bananas in two lots A and B. For A: Rs 2 for 3 bananas; for B: Re 1 each. Total Rs 400. If he had sold A at Re 1 each and B at Rs 4 for 5 bananas, total would be Rs 460. Find the total number of bananas.