NCERT Solutions Maths Class 10 Exercise 3.2
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.
Solution: (i) Let the number of boys who took part in the
quiz be x and the number of girls who took part in the quiz be y.
According to the first
condition,
x + y = 10 …….. (1)
According to the second
condition,
y = x + 4
⇒ x – y = −4 …….. (2)
For equation x + y = 10, we have the following points which lie on the line.
x 
0 
10 
y 
10 
0 
For
equation x – y = –4, we have the following points which lie on the line.
x 
0 
–4 
y 
4 
0 
We
plot the points for both of the equations to find the solution.
Therefore, the
number of boys who took part in the Mathematics quiz is 3 and the number of
girls who took part in the Mathematics quiz is 7.
(ii) Let the cost of one pencil be Rs x and the
cost of one pen be Rs y.
5x + 7y = 50 ……… (1)
x 
10 
3 
y 
0 
5 
7x + 5y = 46 ……… (2)
For equation 7x + 5y = 46, we have the following points which lie on the line.
x 
8 
3 
y 
–2 
5 
We can clearly see that the intersection point
of two lines is (3, 5).
Therefore,
the cost of one pencil is Rs 3 and the cost of one pen is Rs 5.
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:
(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: (i) 5x − 4y + 8 = 0 …… (1)
7x + 6y – 9 = 0 …… (2)
Comparing equation 5x − 4y + 8 = 0 with a_{1}x + b_{1}y + c_{1} = 0 and 7x + 6y – 9 = 0 with a_{2}x + b_{2}y + c_{2 }= 0, we get
a_{1} = 5,
b_{1} = –4, c_{1}
= 8, a_{2} = 7, b_{2} = 6 and c_{2} = –9
We have, a_{1}/a_{2
}≠ b_{1}/b_{2} because 5/7 ≠ –4/6.
Hence, the
equations have unique solution which means the lines representing them intersect
at a point.
(ii) 9x + 3y + 12 = 0 …… (1)
18x + 6y + 24 = 0
…… (2)
Comparing equation 9x + 3y + 12 = 0 with a_{1}x + b_{1}y + c_{1} = 0 and 18x + 6y + 24 = 0 with a_{2}x + b_{2}y + c_{2 }= 0, we get
a_{1} =
9, b_{1} = 3, c_{1} = 12, a_{2} = 18, b_{2} = 6 and c_{2} = 24
We
have, a_{1}/a_{2 }= b_{1}/b_{2 }= c_{1}/c_{2} because
9/18 = 3/6 = 12/24.
Hence,
the lines are coincident.
(iii) 6x − 3y + 10 = 0 …… (1)
2x – y + 9 = 0
…… (2)
Comparing equation 6x − 3y + 10 = 0 with a_{1}x + b_{1}y + c_{1} = 0 and 2x – y + 9 = 0 with a_{2}x + b_{2}y + c_{2 }= 0, we get
a_{1} =
6, b_{1} = –3, c_{1} =
10, a_{2} = 2, b_{2} = –1 and c_{2} = 9
We
have, a_{1}/a_{2 }= b_{1}/b_{2 }≠ c_{1}/c_{2} because
6/2 = –3/–1 ≠ 10/9.
Hence,
the lines are parallel to each other.
3. On comparing the ratios a_{1}/a_{2},
b_{1}/b_{2} and 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) (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
Solution: (i) 3x + 2y = 5 …… (1)
2x − 3y = 7 …… (2)
Comparing equation 3x + 2y = 5 with a_{1}x + b_{1}y + c_{1} = 0 and 2x − 3y = 7 with a_{2}x + b_{2}y + c_{2 }= 0, we get
a_{1}
= 3, b_{1} = 2, c_{1} = −5, a_{2} = 2, b_{2} = −3 and c_{2}
= −7
We
have, a_{1}/a_{2 }= 3/2 and b_{1}/b_{2 }=
2/−3
Here,
a_{1}/a_{2 }≠ b_{1}/b_{2} which means
equations have a unique solution.
Hence,
the given pair of linear equations are consistent.
(ii) 2x − 3y = 8 …… (1)
4x − 6y = 9 …… (2)
Comparing equation 2x − 3y = 8 with a_{1}x + b_{1}y + c_{1} = 0 and 4x − 6y = 9 with
a_{2}x + b_{2}y + c_{2 }=
0, we get
a_{1}
= 2, b_{1} = −3, c_{1} = −8, a_{2} = 4, b_{2} = −6 and c_{2} = −9
Here,
a_{1}/a_{2 }= b_{1}/b_{2 }≠ c_{1}/c_{2} because 1/2_{ }=
1/2_{ }≠ −8/−9
Therefore,
the equations have no solution because they are parallel.
Hence,
the given pair of linear equations are inconsistent.
(iii) (3/2)x + (5/3)y = 7 ……
(1)
9x − 10y = 14 …… (2)
Comparing equation (3/2)x + (5/3)y = 7 with a_{1}x + b_{1}y + c_{1} = 0 and 9x − 10y = 14 with a_{2}x + b_{2}y + c_{2 }= 0, we get
a_{1} = 3/2,
b_{1} = 5/3, c_{1} = −7,
a_{2} = 9, b_{2}
= −10 and c_{2} = −14
We have, a_{1}/a_{2
}= 1/6 and b_{1}/b_{2 }= −1/6
Here,
a_{1}/a_{2 }≠ b_{1}/b_{2} which means
equations have a unique solution.
Hence,
the given pair of linear equations are consistent.
(iv) 5x − 3y = 11 …… (1)
−10x + 6y = −22 ……
(2)
Comparing equation 5x − 3y = 11 with a_{1}x + b_{1}y + c_{1} = 0 and −10x + 6y = −22 with a_{2}x + b_{2}y + c_{2} = 0, we get
a_{1}
= 5, b_{1} = −3, c_{1} = −11, a_{2} =
−10, b_{2} = 6 and c_{2}
= 22
We
have, a_{1}/a_{2 }= −1/2, b_{1}/b_{2 }= −1/2 and
c_{1}/c_{2 }= −1/2
Here, a_{1}/a_{2 }= b_{1}/b_{2 }= c_{1}/c_{2}
Therefore,
the lines are coincident and they have infinitely many solutions.
Hence,
the given pair of linear equations are consistent.
(v) (4/3)x + 2y
= 8 …… (1)
2x + 3y = 12 …… (2)
Comparing equation (4/3)x + 2y = 8 with a_{1}x + b_{1}y + c_{1} = 0 and 2x + 3y = 12 with a_{2}x + b_{2}y + c_{2} = 0, we get
a_{1}
= 4/3, b_{1} = 2, c_{1} = −8, a_{2} = 2, b_{2} = 3 and c_{2} = −12
We
have, a_{1}/a_{2 }= 2/3, b_{1}/b_{2 }= 2/3 and
c_{1}/c_{2 }= 2/3
Here, a_{1}/a_{2 }= b_{1}/b_{2 }= c_{1}/c_{2}
Therefore,
the lines are coincident and they have infinitely many solutions.
Hence,
the given pair of linear equations are consistent.
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 …… (1)
2x + 2y = 10 ……
(2)
Comparing
equation x + y = 5 with a_{1}x + b_{1}y + c_{1} =
0 and 2x + 2y = 10 with a_{2}x + b_{2}y + c_{2}
= 0, we get
a_{1}
= 1, b_{1} = 1, c_{1} = −5, a_{2} = 2, b_{2} = 2 and c_{2} = −10
We
have, a_{1}/a_{2 }= 1/2, b_{1}/b_{2 }= 1/2 and
c_{1}/c_{2 }= 1/2
Here, a_{1}/a_{2 }= b_{1}/b_{2 }= c_{1}/c_{2}
Therefore,
the lines are coincident and they have infinitely many solutions.
Hence,
the given pair of linear equations are consistent.
For
equation x + y = 5, we have following points which lie on the line.
x 
0 
5 
y 
5 
0 
For equation 2x + 2y = 10, we have following points which lie on the line.
x 
1 
2 
y 
4 
3 
We plot the points for both of the equations to find the solution.
Any point which lies on one line also lies on
the other.
We
can take any random value for y and find the corresponding value of x using the
given equation. All such points will lie on both lines and there will be
infinite number of solutions for these equations.
(ii) x – y = 8 …….. (1)
3x − 3y = 16
…….. (2)
Comparing equation x − y = 8 with a_{1}x + b_{1}y + c_{1} = 0 and 3x − 3y = 16 with a_{2}x + b_{2}y + c_{2} = 0, we get
a_{1}
= 1, b_{1} = −1, c_{1} = −8, a_{2} = 3, b_{2} = −3 and c_{2} = −16
We
have, a_{1}/a_{2 }= 1/3, b_{1}/b_{2 }= 1/3 and
c_{1}/c_{2 }= 1/2
Here, a_{1}/a_{2 }= b_{1}/b_{2 }≠ c_{1}/c_{2}
Therefore,
the lines are parallel and they have no solution.
Hence,
the given pair of linear equations are inconsistent.
(iii) 2x + y – 6 = 0 ….. (1)
4x − 2y – 4 = 0
….. (2)
Comparing equation 2x + y – 6 = 0 with a_{1}x + b_{1}y + c_{1} = 0 and 4x − 2y – 4 = 0 with a_{2}x + b_{2}y + c_{2} = 0, we get
a_{1}
= 2, b_{1} = 1, c_{1} = −6, a_{2} = 4, b_{2} = −2 and c_{2} = −4
We
have, a_{1}/a_{2 }= 1/2 and b_{1}/b_{2 }= −1/2
Here, a_{1}/a_{2 }≠ b_{1}/b_{2}
Therefore,
the lines intersect at a point and they have a unique solution.
Hence,
the given pair of linear equations are consistent.
For
equation 2x + y – 6 = 0, we have the following points which lie on the line.
x 
0 
2 
3 
y 
6 
2 
0 
For equation 4x – 2y – 4 = 0, we have the following points which lie on the line.
x 
0 
2 
1 
y 
–2 
2

0 
We plot the points for both of the equations to find the solution.
We
can clearly see that lines are intersecting at (2, 2).
Hence,
x = 2 and y = 2 are the solution of the given linear equations.
(iv) 2x − 2y – 2 = 0 ….. (1)
4x − 4y – 5 = 0
….. (2)
Comparing equation 2x − 2y – 8 = 0 with a_{1}x + b_{1}y + c_{1} = 0 and 4x − 4y – 5 = 0 with a_{2}x + b_{2}y + c_{2} = 0, we get
a_{1}
= 2, b_{1} = −2, c_{1} = −2, a_{2} = 4, b_{2} = −4 and c_{2} = −5
We
have, a_{1}/a_{2 }= 1/2, b_{1}/b_{2 }= 1/2 and
c_{1}/c_{2 }= 2/5
Here, a_{1}/a_{2 }= b_{1}/b_{2 }≠ c_{1}/c_{2}
Therefore,
the lines are parallel and they have no solution.
Hence,
the given pair of linear equations are inconsistent.
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 length of the rectangular garden be x metres and the width of the rectangular garden be y metres.
According
to the first condition, half perimeter = 36 m.
⇒ x
+ y = 36 ……(i)
According
to the second condition, x = y + 4
⇒ x – y = 4 ……..(ii)
Adding
equations (i) and (ii), we get
2x
= 40
⇒ x = 20 m
Putting
the value of x in equation (i), we get
20
+ y = 36
⇒ y = 16 m
Hence,
the length of the rectangular garden is 20 m and its width is 16 m.
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: (i) Let the second linear equation
be a_{2}x + b_{2}y + c_{2 }= 0.
Comparing
the given linear equation 2x +
3y – 8 = 0 with a_{1}x + b_{1}y + c_{1}
= 0, we get
a_{1}
= 2, b_{1} = 3 and c_{1} = –8
We
know that, two lines intersect with each other if a_{1}/a_{2 }≠
b_{1}/b_{2}
So, the second equation can be 3x + 2y = 5 because a_{1}/a_{2 }≠ b_{1}/b_{2}
(ii) Let the second linear equation be a_{2}x
+ b_{2}y + c_{2 }= 0.
Comparing
the given linear equation 2x +
3y – 8 = 0 with a_{1}x + b_{1}y + c_{1}
= 0, we get
a_{1}
= 2, b_{1} = 3 and c_{1} = –8
We know that, two lines are parallel with each other if a_{1}/a_{2 }= b_{1}/b_{2 }≠ c_{1}/c_{2}
So, the second equation can be 4x + 6y = 3 because a_{1}/a_{2 }= b_{1}/b_{2 }≠ c_{1}/c_{2}
(iii) Let the second linear equation be a_{2}x
+ b_{2}y + c_{2 }= 0.
Comparing
the given linear equation 2x +
3y – 8 = 0 with a_{1}x + b_{1}y + c_{1}
= 0, we get
a_{1}
= 2, b_{1} = 3 and c_{1} = –8
We know that, two lines coincide with each other if a_{1}/a_{2 }= b_{1}/b_{2 }= c_{1}/c_{2}
So, the second equation can be 6x + 9y = 24 because a_{1}/a_{2 }= b_{1}/b_{2 }= c_{1}/c_{2}
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 xaxis, and shade the triangular region.
Solution:
For equation x –
y + 1 = 0, we have the following points which lie on the line.
x 
0 
–1 
2 
y 
1 
0 
3 
For equation 3x + 2y – 12 = 0, we have the following points which lie on the line.
x 
4 
0 
2 
y 
0 
6 
3 
Now, let us plot these points to find the two lines.
We can observe
from the above graphs that points of intersection of the lines with the xaxis
are (–1, 0), (2, 3) and (4, 0). Triangle PBC is shaded in the above graph.