Home Notes MCQ's Qestions NCERT Qestions Worksheets Blogs

Chapter 1:Real Numbers

Chapter 2:Polynomials

Chapter 3:Pair of Linear Equations in Two Variables

Chapter 4:Quadratic Equations

Chapter 5: Arithmetic Progression

Chapter 6: Triangles

Chapter 7:Coordinate Geometry

Chapter 8: Introduction to Trigonometry

Chapter 9:Some Applications of Trigonometry

Chapter 10: Circles

Chapter 11:Constructions

Chapter 12:Area Related to Circles

Chapter 13:Surface Areas and Volumes

Chapter 14:Statistics

Chapter 15:Probability

NCERT Solutions for class 10 maths chapter 3

Pair of Linear Equations in Two Variables

NCERT Solutions for class 10 maths chapter 3 Exercise 3.2

Question 1
Form the pair of linear equations in the following problems, and find their solutions graphically.
(i) 10 students of Class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.
(ii) 5 pencils and 7 pens together cost 50, whereas 7 pencils and 5 pens together cost 46. Find the cost of one pencil and that of one pen.
Answer

(i)Let the number of girls is x and number of boys is y
As per the given question
x +y = 10 .......(i)
x= 4 + y
x - y = 4 --------(ii)
Now, for x+y = 10 the solutions are;
Let x=0
0+y=10
y=10
Coordinate (0,10)
Similarly
Let y=0
x + 0 =10
x=10
Coordinate (10,0)
For x = 4 + y, the solutions are;
let x = 0
0 = 4 + y
y = -4
coordinate (0, -4)
Similarly
let y=0 x = 4 + 0
x=4
Coordinate (4,0)

NCERT Ex-3.2 class 10

From the graph, lines intersecting each other at point (7, 3). So there are 7 girls and 3 boys in the class.


(ii) Let costs of 1 pencil is Rs.x and costs 1 pen is Rs.y.
According to the question,
5x + 7y = 50
7x + 5y = 46
For, 5x + 7y = 50 the solutions are;
Let x = 0
5 (0) + 7y =50
y = 50 /7
Coordinate (0,50/7)
Similarly
Let y=0
5x + 7(0) =50
x = 50/5
x = 10
Coordinate (10,0)

For 7x + 5y = 46 the solutions are;
Let x=0
7(0)+ 5y=46
y = 46 /5 or 9.2
Coordinate (0,46/5)
Similarly
Let y=0
7x + 5(0)= 46
x = 46 /7 or 6.2
Coordinate (46/7 , 0)

Hence, the graphical representation is as follows;
NCERT Ex-3.2 class 10

From the graph, lines cross each other at point (3, 5).
So, the cost of a pencil is 3/- and cost of a pen is 5/-.
Question 2
On comparing the ratios a1/a2 , b1/b2 , c1/c2 find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:
(i) 5x – 4y + 8 = 0
7x + 6y – 9 = 0
(ii) 9x + 3y + 12 = 0
18x + 6y + 24 = 0
(iii) 6x – 3y + 10 = 0
2x – y + 9 = 0
Answer
(i) Given expressions;
5x−4y+8 = 0
7x+6y−9 = 0
Comparing these equations with a1x+b1y+c1 = 0
And a2x+b2y+c2 = 0
We get,
a1 = 5, b1 = -4, c1 = 8
a2 = 7, b2 = 6, c2 = -9
(a1/a2) = 5/7
(b1/b2) = -4/6 = -2/3
(c1/c2) = 8/-9
Since, (a1/a2) ≠ (b1/b2)
So, the pairs of equations have a unique solution

(ii) Given expressions;
9x + 3y + 12 = 0
18x + 6y + 24 = 0
Comparing these equations with a1x+b1y+c1 = 0
And a2x+b2y+c2 = 0
We get,
a1 = 9, b1 = 3, c1 = 12
a2 = 18, b2 = 6, c2 = 24
(a1/a2) = 9/18 = 1/2
(b1/b2) = 3/6 = 1/2
(c1/c2) = 12/24 = 1/2
Since (a1/a2) = (b1/b2) = (c1/c2)
So, the pairs of equations have infinite possible solutions and the lines are coincident.

(iii) Given Expressions;
6x – 3y + 10 = 0
2x – y + 9 = 0
Comparing these equations with a1x+b1y+c1 = 0
And a2x+b2y+c2 = 0

We get,
a1 = 6, b1 = -3, c1 = 10
a2 = 2, b2 = -1, c2 = 9
(a1/a2) = 6/2 = 3/1
(b1/b2) = -3/-1 = 3/1
(c1/c2) = 10/9
Since (a1/a2) = (b1/b2) ≠ (c1/c2)
So, the pairs of equations are parallel to each other so there is no possible solution for the given pair of equations.

Question 3
On comparing the ratio, (a1/a2) , (b1/b2) , (c1/c2) find out whether the following pair of linear equations are consistent, or inconsistent.
(i) 3x + 2y = 5 ; 2x – 3y = 7
(ii) 2x – 3y = 8 ; 4x – 6y = 9
(iii)(3/2)x+(5/3)y = 7; 9x – 10y = 14
(iv) 5x – 3y = 11 ; – 10x + 6y = –22
(v)(4/3)x+2y = 8 ; 2x + 3y = 12
Answer

(i) Given : 3x + 2y = 5 or 3x + 2y -5 = 0
and 2x – 3y = 7 or 2x – 3y -7 = 0
Comparing these equations with a1x+b1y+c1 = 0
And a1x+b1y+c1 = 0
We get,
a1 = 3, b1 = 2, c1 = -5
a2 = 2, b2 = -3, c2 = -7
(a1/a2) = 3/2
(b1/b2) = 2/-3
(c1/c2) = -5/-7 = 5/7
Since, (a1/a2) ≠ (b1/b2)
So, the given equations intersect each other at one point and have only one possible solution. The equations are consistent.

(ii) Given 2x – 3y = 8 and 4x – 6y = 9
Therefore,
a1 = 2, b1 = -3, c1 = -8
a2 = 4, b2 = -6, c2 = -9
(a1/a2) = 2/4 = 1/2
(b1/b2) = -3/-6 = 1/2
(c1/c2) = -8/-9 = 8/9
Since , (a1/a2) = (b1/b2) ≠ (c1/c2)
So, the equations are parallel to each other and they have no possible solution. Hence, the equations are inconsistent.

(iii)Given (3/2)x + (5/3)y = 7 and 9x – 10y = 14
Therefore,
a1 = 3/2, b1 = 5/3, c1 = -7
a2 = 9, b2 = -10, c2 = -14
(a1/a2) = 3/(2×9) = 1/6
(b1/b2) = 5/(3× -10)= -1/6
(c1/c2) = -7/-14 = 1/2
Since, (a1/a2) ≠ (b1/b2)
So, the equations are intersecting each other at one point and they have only one possible solution. Hence, the equations are consistent.

(iv) Given, 5x – 3y = 11 and – 10x + 6y = –22
Therefore,
a1 = 5, b1 = -3, c1 = -11
a2 = -10, b2 = 6, c2 = 22
(a1/a2) = 5/(-10) = -5/10 = -1/2
(b1/b2) = -3/6 = -1/2
(c1/c2) = -11/22 = -1/2
Since (a1/a2) = (b1/b2) = (c1/c2)
Infinite number of possible solutions. Hence, the equations are consistent.

(v)Given, (4/3)x +2y = 8 and 2x + 3y = 12
a1 = 4/3 , b1 = 2 , c1 = -8
a2 = 2, b2 = 3 , c2 = -12
(a1/a2) = 4/(3×2)= 4/6 = 2/3
(b1/b2) = 2/3
(c1/c2) = -8/-12 = 2/3
Since (a1/a2) = (b1/b2) = (c1/c2)
Infinite number of possible solutions. Hence, the equations are consistent.
Question 4
Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:
(i) x + y = 5, 2x + 2y = 10
(ii) x – y = 8, 3x – 3y = 16
(iii) 2x + y – 6 = 0, 4x – 2y – 4 = 0
(iv) 2x – 2y – 2 = 0, 4x – 4y – 5 = 0
Answer

(i)Given, x + y = 5 and 2x + 2y = 10
(a1/a2) = 1/2
(b1/b2) = 1/2
(c1/c2) = 1/2
Since (a1/a2) = (b1b2) = (c1/c2)
∴The equations are coincident and they have infinite number of possible solutions.
So, the equations are consistent.
For, x + y = 5
Let x=0
0+ y=5
y=5
Co-ordinate (0,5)
Similarly
Let y=0
x+ 0 =5
x=5
Co-ordinate (5,0)
For 2x + 2y = 10
2(x+y)= 10
x+y= 5
Let x=0
0+ y=5
y=5
co-ordinate (0,5)
Similarly
Let y=0
x+ 0 =5
x=5
co-ordinate (5,0)
NCERT Ex-3.2 class 10


The lines are overlapping each other.Therefore, It have infinite possible solutions.

(ii) Given, x – y = 8 and 3x – 3y = 16
(a1/a2) = 1/3
(b1/b2) = -1/-3 = 1/3
(c1/c2) = 8/16 = 1/2
Since, (a1/a2) = (b1/b2) ≠ (c1/c2)
The equations are parallel to each other and have no solutions. Hence, the pair of linear equations is inconsistent.

(iii) Given, 2x + y – 6 = 0 and 4x – 2y – 4 = 0
(a1/a2) = 2/4 = 1/2
(b1/b2) = 1/-2
(c1/c2) = -6/-4 = 3/2
Since, (a1/a2) ≠ (b1/b2)
The given linear equations are intersecting each other at one point and have only one solution. Hence, the pair of linear equations is consistent.
For 2x + y – 6 = 0
2x + y =6
Let x=0
2(0) + y = 6
y=6
Co-ordinate (0,6)
Similarly
Let y =0
2x + 0 =6
x= 3
Co-ordinate (3,0)
And for 4x – 2y – 4 = 0
2( 2x - y) = 4
2x - y = 4
Let x =0
2(0) - y = 4
-y = 4
Co-ordinate (0,-4)
Similarly
Let y=0
2x - 0 =4
2x = 4
x= 2
Co-ordinate (2,0)
NCERT Ex-3.2 class 10

The lines are intersecting each at point,(2,2).

(iv) Given, 2x – 2y – 2 = 0 and 4x – 4y – 5 = 0
(a1/a2) = 2/4 = ½
(b1/b2) = -2/-4 = 1/2
(c1/c2) = 2/5
Since, a1/a2 = b1/b2 ≠ c1/c2
Thus, these linear equations have parallel and have no possible solutions. Hence, the pair of linear equations are inconsistent.

Question 5
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.
Answer
Let the width of the garden is x and length is y.
Now, according to the question
y = 4 + x
y + x = 36
Now, taking y = x + 4
Let x = 0
y = 0 +4
y=4
Co-ordinate (0,4)
Similarly
Let y=0
0 = x + 4
x = -4
Co-ordinate (-4,0)
For y + x = 36
Let x=0
y + 0 =36
y = 36
Co-ordinate (0,36)
Similarly
Let y=0
0 + x = 36
x= 36
Co-ordinate (36,0)

NCERT Ex-3.2 class 10

The lines intersects each other at a point(16, 20)
width of the garden is 16
length is 20.
Question 6
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
Answer

(i) Given the linear equation 2x + 3y – 8 = 0.
For Intersecting lines

(a1/a2) ≠ (b1/b2)
Thus, another equation could be 3x – 7y + 9 = 0, such that;
(a1/a2) = 2/3 and (b1/b2) = 3/-7
Clearly, equation satisfies the condition.

(ii) Given the linear equation 2x + 3y – 8 = 0.
For Parallel lines
(a1/a2) = (b1/b2) ≠ (c1/c2)
Thus, another equation could be 8x + 12y + 9 = 0, such that;
(a1/a2) = 2/8 = 1/4
(b1/b2) = 3/12= 1/4
(c1/c2) = -8/9
Clearly, equation satisfies the condition.

(iii) Given the linear equation 2x + 3y – 8 = 0.
For Coincident lines
(a1/a2) = (b1/b2) = (c1/c2)
Thus, another equation could be 8x + 12y – 24 = 0, such that;
(a1/a2) = 2/8 = 1/4 ,(b1/b2) = 3/12 = 1/4, (c1/c2) = -8/-24 = 1/4
Clearly, equation satisfies the condition.
Question 7
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.
Answer
Given, the equations for graphs are x – y + 1 = 0 and 3x + 2y – 12 = 0.
For, x – y + 1 = 0
Let x=0
0 - y = -1
y = 1
Co-ordinate (0,1)
Similarly
Let y=0
x - 0 = -1
x = -1
Co-ordinate (-1,0)
For, 3x + 2y – 12 = 0
let x =0
3(0) + 2y =12
2y = 12
y = 6
Co-ordinate (0,6)
Similarly
Let y=0
3x + 2(0) = 12
3x = 12
x = 4
Co-ordinate (4,0)
NCERT Ex-3.2 class 10


The lines are intersecting each other at point (2, 3).