Class 10 Mathematics

# Linear Equations

## Exercise 3.2 (NCERT) Part 1

Question 1: Form a pair of linear equations in the following problems, and find their solutions graphically.

(a) 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.

Solution: Let us assume that number of boys = x and number of girls = y. We get following equations as per question:

x + y = 10
Or, y = 10 – x ………(1)

This equation will give following values for x and y;

 x 1 2 3 4 y 9 8 7 6

y = x + 4 ………..(2)

This equation will give following values for x and y;

 x 1 2 3 4 y 5 6 7 8

Following graph is plotted for the given pair of linear equations:

Number of boys = 3 and number of girls = 7

(b) 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: Let us assume that price of a pencil is x and that of a pen is y. We get following equations as per question:

5x + 7y = 50 ………(1)

This equation will give following values for x and y;

 x 1 2 3 4 y 6.4 5.7 5 4.2

7x + 5y = 46 ……….(2)

This equation will give following values for x and y;

 x 1 2 3 4 y 7.8 6.4 5 3.6

Following graph is plotted for the given pair of linear equations.

Price of one pencil = Rs. 3 and Price of one pen = Rs. 5

Question 2: On comparing the ratios (a_1)/(a_2), (b_1)/(b_2) and (c_1)/(c_2) find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident.

(a) 5x – 4y + 8 = 0 and 7x + 6y – 9 = 0

Solution: In the given pair of linear equations;

(a_1)/(a_2)=5/7

(b_1)/(b_2)=-4/6=-2/3

It is clear that;

(a_1)/(a_2)≠(b_1)/(b_2)

Hence the lines representing the given pair of linear equations intersect at a point.

(b) 9x + 3y + 12 = 0 and 18x + 6y + 24 = 0

Solution: In the given pair of linear equations;

(a_1)/(a_2)=(9)/(18)=1/2

(b_1)/(b_2)=3/6=1/2

(c_1)/(c_2)=(12)/(24)=1/2

It is clear that;

(a_1)/(a_2)=(b_1)/(b_2)=(c_1)/(c_2)

Hence the lines representing the given pair of linear equations will be coincident.

(c) 6x – 3y + 10 = 0 and 2x- y + 9 = 0

Solution: For the given pair of linear equations;

(a_1)/(a_2)=6/2=3

(b_1)/(b_2)=(-3)/(-1)=3

(c_1)/(c_2)=(10)/(9)

It is clear that;

(a_1)/(a_2)=(b_1)/(b_2)≠(c_1)/(c_2)

Hence the lines representing the given pair of linear equations will be parallel.

Question 3: On comparing the ratios (a_1)/(a_2), (b_1)/(b_2) and (c_1)/(c_2) find out whether the following pairs of linear equations are consistent or inconsistent.

(a) 3x + 2y = 5 and 2x – 3y = 7

Solution: For the given pair of linear equations;

(a_1)/(a_2)=3/2

(b_1)/(b_2)=(2)/(-3)

It is clear that;

(a_1)/(a_2)≠(b_1)/(b_2)

Hence, the given pair of linear equations is consistent.

(b) 2x – 3y = 8 and 4x – 6y = 9

Solution: For the given pair of linear equations;

(a_1)/(a_2)=2/4=1/2

(b_1)/(b_2)=(-3)/(-6)=1/2

(c_1)/(c_2)=8/9

It is clear that;

(a_1)/(a_2)=(b_1)/(b_2)≠(c_1)/(c_2)

Hence the given pair of linear equations is inconsistent.

(c) (3)/(2)x + (5)/(3)y = 7 and 9x – 10y = 14

Solution: For the given pair of linear equations;

(a_1)/(a_2)=(3)/(2)÷9=(3)/(18)

(b_1)/(b_2)=(5)/(3)÷-10=-5/6

It is clear that;

(a_1)/(a_2)≠(b_1)/(b_2)

(d) (4)/(3)x + 2y = 8 and 2x + 3y = 12

Solution: For the given pair of linear equations;

(a_1)/(a_2)=(4)/(3)÷2=(2)/(3)

(b_1)/(b_2)=2/3

(c_1)/(c_2)=2/3

It is clear that;

(a_1)/(a_2)=(b_1)/(b_2)= (c_1)/(c_2)

Hence the given pair of linear equations is dependent and consistent.

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Introduction

Exercise:3.1

Exercise:3.2(Part 2)

Exercise:3.3(Part 1)

Exercise:3.3 (part 2)

Exercise:3.4 (part 1)

Exercise:3.4 (part 2)

Exercise:3.5 (part 1)

Exercise:3.5 (part 2)

Exercise:3.6 (part 1)

Exercise:3.6 (part 2)

Exercise:3.6 (part 3)