NCERT SOLUTIONS FOR CLASS 10 MATHS PAIR OF LINEAR EQUATIONS IN TWO VARIABLES CHAPTER 3 Ex 3.2

NCERT SOLUTIONS FOR CLASS 10 MATHS PAIR OF LINEAR EQUATIONS IN TWO VARIABLES CHAPTER 3 EX 3.2

Question 1. Form the pair of linear equations of 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 Rs 50, whereas 7 pencils and 5 pens together cost Rs 46. Find the cost of one pencil and that of one pen.
Solution:

(i) Let the number of girls be x and number of boys be y.

According to question,

1st Condition:

x + y = 10

Table: made to find the values of Variables x and y

x465
y645

2nd Condition:

x = y + 4 ⇒ x – y = 4

Table: made to find the values of Variables x and y

x867
y423

Solving

Represent 1st Condition and 2nd Condition in the graph

(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.

Solution:

(i) Let the number of Pencils be x and the number of Pens be y.

According to question,

1st Condition:

5x + 7y = 50

Table: made to find the values of Variables x and y

x3 7 4
y52.14.3

2nd Condition:

7x + 5y = 46

Table: made to find the values of Variables x and y

x 5 3 2
y2.2 56.2

Question 2. On comparing the ratios a1a2, b1b2
and c1c2 , 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
Solution:

Hence, the lines are coincident.

Hence, the pair of linear equations has no solution, i.e the lines are parallel.

Question 3. On comparing the ratios a1a2, b1b2
and c1c2, find out whether the following pairs of linear equations are consistent, or inconsistent:
Solution:

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
Solution:

(i) x + y – 5, 2x + 2y = 10

System of equations is inconsistent and the graph represents coincident lines.

1st Condition:

Table: made to find the values of Variables x and y

x1423
y4132

2nd Condition:

Table: made to find the values of Variables x and y

x1423
y4132

Pair of equations is inconsistent.

Hence, lines are parallel

(iii) 2x + y – 6 = 0, 4x – 2y – 4 = 0

Pair of equations is consistent.

1st Condition:

Table: made to find the values of Variables x and y for equation 2x + y – 6 = 0

x21-1
y248

2nd Condition:

Table: made to find the values of Variables x and y for equation 4x – 2y – 4 = 0

x102
y0-22

Pair of equations is inconsistent. Hence, lines are parallel and system has no solution.

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 graphically.
Solution:

Let us consider.
The width of the garden is x and length is y.
Now, according to the question, we can express the given condition as;
y – x = 4

and
y + x = 36
Now, taking y – x = 4 or y = x + 4
For y + x = 36, y = 36 – x
The graphical representation of both the equation is as follows;
From the graph you can see, the lines intersect each other at a point(16, 20). Hence, the
width of the garden is 16 and the 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
Solution:

Given equation is 2x + 3y – 8 = 0

we have 2x + 3y = 8

Let required equation be ax + by = c

Condition:

(i) For intersecting lines

Where a, b, c can have any value which satisfies the above condition.

Equations are 2x + 3y = 8 and 3x + 2y = 4 have unique solution and their geometrical representation shows intersecting lines.

(ii) Given equation is 2x + 3y = 8 required equation be ax + by = c

Condition:

For parallel lines

Where a, b, c can have any value which satisfy the above condition.

Let, a = 2, b = 3, c = 4

Required equation will be 2x + 3y = 4

Equation 2x + 3y = 8 and 2x + 3y = 4 have no solution and their geometrical representation shows parallel lines.

(iii) For Coincident lines:

Given equation is 2x + 3y = 8.

Let required equation be ax + by = c For coincident lines.

Where a, b, c can have any a, b, c possible value that satisfies the above condition.

Let a = 4, b= 6, c = 16

Required equation will be 4x + 6y = 16

Equations 2x + 3y = 8 and 4x + 6y = 16 have infinitely many solutions and their geometrical representation shows coincident lines.

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.
Solution:

Thus, we have the following table of solutions:

x024
y630

Plotting the points in a graph paper, we get the shaded triangle ABC with vertices A(2, 3), B(-1, 0) and C(4, 0)

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