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 Rs 50, whereas 7 pencils and 5 pens together cost Rs. 46. Find the cost of one pencil and that of one pen.

Answer:

(i) Let number of boys = x

Let number of girls = y

According to given conditions, we have

x + y = 10

And, x = 10 – y

putting y=0,5,10,we get,
X=10-0=10
X=10-5=5,
X=10-10=0

x	10	5	0
y	0	5	10

Number of girls is 4 more than number of boys ……..Given,
so,
Y=x+4
putting x=-4,0,4 we get,
Y=-4+4=0
Y=0+4
Y=4+4=8

x	-4	0	4
y	0	4	8

We plot the points for both of the equations to find the solution.

(ii)

Let the cost of one pencil=Rs.X
and Let the cost of one pen=Rs.Y
According to the given conditions, we have:
=5x + 7y = 50
=5x=50-7y
=x=10-7/5y

Three solutions of this equation can be written in a table as follows:

x	3	-4	-11
y	5	10	15

Three solutions of this equation can be written in a table as follows:

x	0	2	4
y	9.2	6.4	3.6

The graphical representation is as follows:

 

2. On comparing the ratios a₁/a₂,b₁/b₂ and 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

(ii)9x + 3y + 12 = 0

7x + 6y – 9 = 018x + 6y + 24 = 0

(iii) 6x − 3y + 10 = 0

2x – y + 9 = 0

Answer:

(i) 5x − 4y + 8 = 0, 7x + 6y – 9 = 0

Comparing equation 5x − 4y + 8 = 0 with a₁x + b₁y + c₁ = 0and 7x + 6y – 9 = 0 with a₂x + b₂y + c₂ = 0,

We get,

Hence,

we find that,

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

Comparing equation 9x + 3y + 12 = 0 with a₁x + b₁y + c₁ = 0and 7x + 6y – 9 = 0 with a₂x + b₂y + c₂ = 0,

We get,

Hence

We find that,

Hence, lines are coincident.

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

Comparing equation 6x − 3y + 10 = 0 with a₁x + b₁y + c₁ = 0and 7x + 6y – 9 = 0 with a₂x + b₂y + c₂ = 0,

We get, 

Hence

We find that,

Hence,
lines are parallel to each other.

3. On comparing the ratios a₁/a₂,b₁/b₂ and c₁/c₂ , find out whether the following pair of linear equations are consistent, or inconsistent.

(i) 3x + 2y = 5, 2x − 3y = 8

(ii) 2x − 3y = 7, 4x − 6y = 9

(iii) 3x/2 + 5y/3 = 7, 9x − 10y = 14

(iv) 5x − 3y = 11, −10x + 6y = −22

Answer:

(i) 3x + 2y = 5, 2x − 3y = 7

Comparing equation 3x + 2y = 5 with a₁x + b₁y + c₁ = 0and 7x + 6y – 9 = 0 with a₂x + b₂y + c₂ = 0,

We get,

Hence,

Therefore,these linear equations will intersect at one point only and have only one possible solution.
And,pair of linear euations is consistent

(ii) 2x − 3y = 8, 4x − 6y = 9

Comparing equation 2x − 3y = 8 with a₁x + b₁y + c₁ = 0and 7x + 6y – 9 = 0 with a₂x + b₂y + c₂ = 0,

We get,

Hence,

Therefore,these linear equations are parallel to each other and have no possible solution.in
And,pair of linear euations is inconsistent

(iii) 9x − 10y = 14

We get,

Hence,

Therefore, these linear equations will intersect each other at one point and have only one possible solution.
And,pair of linear euations is consistent

(iv) 5x − 3y = 11, −10x + 6y = −22

Comparing equation 5x − 3y = 11 with a₁x + b₁y + c₁ = 0and 7x + 6y – 9 = 0 with a₂x + b₂y + c₂ = 0,

We get,

Hence,

Therefore these pair of lines have infinite number of solutions
And,pair of linear euations is consistent

4. Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:

4. (i) x + y = 5, 2x + 2y = 10

(ii) x – y = 8, 3x − 3y = 16

(iii) 2x + y = 6, 4x − 2y = 4

(iv) 2x − 2y – 2 = 0, 4x − 4y – 5 = 0

Answer:

(i) x + y = 5, 2x + 2y = 10

We get,

Hence,

(ii) x – y = 8, 3x − 3y = 16

We get,

Hence,

Therefore, these linear equations are intersecting each other at one point and thus have only one possible solution.
Hence,the pair of linear equations is inconsistent.

(iii) 2x + y = 6, 4x − 2y = 4

We get,

Hence,

Therefore, these linear equations are intersecting each other at one point and thus have only one possible solution.
Hence,the pair of linear equations is consistent

(iv) 2x − 2y – 2 = 0, 4x − 4y – 5 = 0

We get,

Hence,

Therefore, these linear equations are parallel to each other and have no possible solution,
Hence,the pair of linear equations is inconsistent.

5. Half the perimeter of a rectangle garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.

Answer:

Let width of rectangular garden = x metres
and length=y

So,

Hence, the graphic representation is as follows.

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) Let the second line be equal to a₂x + b₂y + c₂ = 0,

Intersecting Lines:For this Condition,

The Second line such that it is intersecting the given line is
2x+4y-6=0

As,

(ii) Let the second line be equal to a₂x + b₂y + c₂ = 0,

parallel Lines:
For this Condition,

Hence,the second line can be 4x+6y-8=0

As,

(iii) Let the second line be equal to a₂x + b₂y + c₂ = 0,

Coincident lines: For coincident lines,

Hence,the second line can be 6x+9y-24=0

As,

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:

For equation x – y + 1 = 0, we have following points which lie on the line

For equation 3x + 2y – 12 = 0, we have following points which lie on the line

We can see from the graphs that points of intersection of the lines with the x–axis are (–1, 0), (2, 3) and (4, 0).

CHAPTER NAME	OLD NCERT	NEW NCERT
Real Numbers	EXERCISE 1.1
	EXERCISE 1.2	1.1	CLICK HERE
	EXERCISE 1.3	1.2	CLICK HERE
	EXERCISE 1.4
Polynomials	EXERCISE 2.1	2.1	CLICK HERE
	EXERCISE 2.2	2.2	CLICK HERE
	EXERCISE 2.3
	EXERCISE 2.4
Pair of Linear Equations in Two Variables	EXERCISE 3.1
	EXERCISE 3.2	3.1	CLICK HERE
	EXERCISE 3.3	3.2	CLICK HERE
	EXERCISE 3.4	3.3	CLICK HERE
	EXERCISE 3.5
	EXERCISE 3.6
	EXERCISE 3.7
Quadratic Equations	EXERCISE 4.1	4.1	CLICK HERE
	EXERCISE 4.2	4.2	CLICK HERE
	EXERCISE 4.3
	EXERCISE 4.4	4.3	CLICK HERE
Arithmetic Progressions	EXERCISE 5.1	5.1	CLICK HERE
	EXERCISE 5.2	5.2	CLICK HERE
	EXERCISE 5.3	5.3	CLICK HERE
	EXERCISE 5.4	5.4 (Optional)	CLICK HERE
Triangles	EXERCISE 6.1	6.1	CLICK HERE
	EXERCISE 6.2	6.2	CLICK HERE
	EXERCISE 6.3	6.3	CLICK HERE
	EXERCISE 6.4
	EXERCISE 6.5
	EXERCISE 6.6
Coordinate Geometry	EXERCISE 7.1	7.1	CLICK HERE
	EXERCISE 7.2	7.2	CLICK HERE
	EXERCISE 7.3
	EXERCISE 7.4
Introduction to Trigonometry	EXERCISE 8.1	8.1	CLICK HERE
	EXERCISE 8.2	8.2	CLICK HERE
	EXERCISE 8.3
	EXERCISE 8.4	8.3	CLICK HERE
Some Applications of Trigonometry	EXERCISE 9.1	9.1	CLICK HERE
Circles	EXERCISE 10.1	10.1	CLICK HERE
	EXERCISE 10.2	10.2	CLICK HERE
Construction
Areas Related to Circles	EXERCISE 12.1
	EXERCISE 12.2	11.1	CLICK HERE
	EXERCISE 12.3
Surface Areas and Volumes	EXERCISE 13.1	12.1	CLICK HERE
	EXERCISE 13.2	12.2	CLICK HERE
	EXERCISE 13.3
	EXERCISE 13.4
	EXERCISE 13.5
Statistics	EXERCISE 14.1	13.1	CLICK HERE
	EXERCISE 14.2	13.2	CLICK HERE
	EXERCISE 14.3	13.3	CLICK HERE
	EXERCISE 14.4
Probability	EXERCISE 15.1	14.1	CLICK HERE
	EXERCISE 15.2