In this article, we will share MP Board Class 10th Maths Book Solutions Pair of Linear Equations in Two Variables Ex 3.2 Pdf, Class 10 Maths Chapter 3 Exercise 3.2 Solutions, These solutions are solved subject experts from the latest edition books.

## MP Board Class 10th Maths Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.2

**3.2 Maths Class 10 MP Board 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.

Solution:

(i) Let the number of boys = x and Number of girls = y

∴ x + y = 10 …. (1)

Also, Number of girls = [Number of boys] + 4

∴ y = x + 4 ⇒ x – y = – 4 … (2)

Now, from the equation (1), we have :

l_{1} : x + y = 10 ⇒ y = 10 – x

And from the equation (2), we have

l_{2} : x – y = -4 ⇒ y = x + 4

The lines l_{1} and l_{2} intersects at the point (3, 7)

∴ The solution of the pair of linear equations is : x = 3, y = 7

⇒ Number of boys = 3

and number of girls = 7

Let the cost of a pencil = ₹ x

and cost of a pen = ₹ y

Since, cost of 5 pencils + Cost of 7 pens = ₹ 50

5 x + 7y = 50 … (1)

Also, cost of 7 pencils + cost of 5 pens = ₹ 46

7x + 5y = 46 …. (2)

Now, from equation (1), we have

And from equation (2), we have

Plotting the points (10, 0), (3, 5) and (-4, 10), we get a straight line l_{1} and plotting the points (8, -2), (3, 5) and (0, 9.2) we get a straight line l_{2}.

These two straight lines intersects at (3, 5).

∴ Cost of a pencil = ₹ 3 and cost of a pen = ₹ 5.

**Class 10 Math Chapter 3.2 Question 2.**

On comparing the ratios \(\frac{a_{1}}{a_{2}}, \frac{b_{1}}{b_{2}}\) and \(\frac{c_{1}}{c_{2}}\) 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:

Comparing the given equations with

a_{1}x + b_{1}y + C_{1} = 0, a_{2}x + b_{2}y + c_{2} = 0, we have

(i) For, 5x – 4y + 8 = 0, 7x + 6y – 9 = 0,

a_{1} = 5, b_{1} = -4, C_{1} = 8; a_{2} = 7, b_{2} = 6, c_{2} = -9

So, the lines are intersecting, i.e., they intersect at a point.

(ii) For, 9x + 3y + 12 = 0,18x + 6y + 24 = 0,

a_{1} = 9, b_{1} = 3, C_{1} = 12; a_{2} = 18, b_{2} = 6, c_{2} = 24

So, the lines are coincident.

(iii) For, 6x – 3y +10 = 0, 2x – y + 9 = 0

So, the lines are parallel.

**Exercise 3.2 Class 10 Maths Question 3.**

On comparing the ratios \(\frac{a_{1}}{a_{2}}, \frac{b_{1}}{b_{2}}\) and \(\frac{c_{1}}{c_{2}}\), 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) \(\frac{3}{2} x+\frac{5}{3} y\) = 7; 9x-10y = 14

(iv) 5x – 3y = 11; -10x + 6y = -22

Solution:

Comparing the given equations with

So, lines are intersecting i.e., they intersect at a point.

∴ It is consistent pair of equations.

(ii) For 2x – 3y = 8, 4x – 6y = 9, 2x – 3y – 8 = 0 and 4x – 6y – 9 = 0

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

∴ It is inconsistent pair of equations.

(iii)

⇒ The given pair of linear equations has exactly one solution.

∴ It is a consistent pair of equations.

(iv)

So, lines are coincident.

⇒ The given pair of linear equations has infinitely many solutions.

Thus, it is consistent pair of equations

(v)

So, the lines are coincident.

⇒ The given pair of linear equations have infinitely many solutions.

Thus it is consistent pair of equations.

**3.2 Maths Class 10 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) For, x + y = 5, 2x + 2y = 10 ⇒ x + y – 5 = 0 and 2x + 2y – 10 = 0

So, lines l_{1} and l_{2} are coinciding. i.e., They have infinitely many solutions.

(ii) For, x – y = 8,

3x – 3y = 16

⇒ x – y – 8 = 0

and 3x – 3y = 16

so, line are parallel

∴ The pair of linear equations are inconsistent

(iii) For 2x + y – 6 = 0, 4x – 2y – 4 = 0

The pair of linear equations are inconsistent

**Consistent And Inconsistent Equations Class 10 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.

Solution:

Let the width of the garden = x m

and the length of the garden = y m

According to question,

The lines l_{1} and l_{2} intersect at (16, 20).

∴ x = 16 and y = 20

So, Length = 20m, and Width = 16m

**Class 10 Maths Chapter 3 Exercise 3.2 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:

Let the pair of linear equations be 2x + 3y – 8 = 0

⇒ a_{1} = 2, b_{1} = 3 and C_{1} = -8 and a_{2}x + b_{2}y + c_{2} = 0.

(i) For intersecting lines, we have:

∴ We can have a_{2} = 3, b_{2} = 2 and c_{2} = – 7

∴ The required equation can be 3x + 2y – 7 = 0

(ii) For parallel lines, \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}\)

∴ Any line parallel to 2x + 3y – 8 = 0, can be taken as 2x + 3y -12 = 0

(iii) For coincident lines, \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)

∴ Any line parallel to 2x + 3y – 8 = 0 can be 2(2x + 3y – 8 = 0)

⇒ 4x + 6y – 16 = 0

**Class 10 Maths Exercise 3.2 Solutions 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:

Since, we have

The lines l_{1} and l_{2} intersect at (2, 3). Thus, co-ordinates of the vertices of the shaded triangular region are (4, 0), (-1, 0) and (2, 3).