Class10 NCRT Pair of Linear Equations in Two Variables Exercise – 3.2 pdf || UP Board
Pair of Linear Equations in Two
Variables
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
₹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
= y
Let the
number of girls = x
Total
number of students = 10
So,
x +
y =
10 ………………..(1)
Now,
according to the question, the number of girls is 4
more than the number of
boys,
x =
y +
4 ……………….(2)
Both
equations need three solutions to be expressed graphically.
From
equation (1),
x + y = 10
x = 10 - y
|
x |
4 |
6 |
5 |
|
y |
6 |
4 |
5 |
From
equation (2),
x = y + 4
|
x |
5 |
3 |
7 |
|
y |
1 |
-1 |
3 |
On looking
at the graph, we find that the point of intersection of the equations is (7, 3)
which is the common
solution of the linear equations.
Hence,
number of girls = 7 and number of boys = 3.
(ii)
Let the cost of one
pencil = ₹
x
Let the
cost of a pen = ₹ y
According
to the first condition,
So,
5x +
7y =
50 ………………..(1)
According
to the second condition,
7x +
5y =
46 ……………….(2)
Both
equations need three solutions to be expressed graphically.
From
equation (1),
5x + 7y =
50
|
x |
10 |
3 |
-4 |
|
y |
0 |
5 |
10 |
From
equation (2),
7x +
5y =
46
|
x |
8 |
-2 |
3 |
|
y |
-2 |
12 |
5 |
On looking
at the graph, we find that the point of intersection of the equations is (3, 5)
which is the common
solution of the linear equations.
Hence,
cost of a pencil = ₹ 3 and cost of a pen = ₹ 5.
2. On
comparing the ratios a1/a2, b1/b2 and c1/c2 find
our 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
Solution:
a1 = 5, b1 = -4, c1 =
8
Hence, the lines represented by the pair of equations
intersect at a point.
(ii)9x + 3y + 12 = 0
18x + 6y +24 = 0
Solution:
a1 = 9, b1 = 3, c1 =
12
a2 = 18, b2 = 6, c2 =
24
Hence pair of linear equations are coincident.
(iii)6x - 3y + 10 = 0
2x - y + 9 = 0
Solution:
a1 = 6, b1 = -3, c1 =
10
Here,
Hence,
pair of linear equations are parallel.
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 = 7
Solution:
a1 = 3, b1 = 2, c1 =
5
Hence, the
pair of linear equations Intersect at one point and the pair of linear
equations are consistent.
(ii)2X - 3Y = 8, 4X - 6Y = 9
Solution:
a1 = 2, b1 = -3, c1 =
8
Hence, the
pair of linear equations are parallel and the pair of linear equations are
inconsistent.
(iii) (3/2)x + (5/3)y = 7, 9x – 10y =
14
Solution:
9x + 10y = 42, 9x – 10y = 14
a1 = 9, b1 = 10, c1 =
42
Hence, the
pair of linear equations intersect at one point and the pair of linear
equations is consistent.
(iv) 5x - 3y = 11, -10x + 6y = -22
Solution:
a1 = 5, b1 = -3, c1 =
11
Hence, the
pair of linear equations are consistent.
(v) (4/3)x + 2y = 8, 2x + 3y = 12
Solution:
a1 = 4/3, b1 = 2, c1
= 8
Hence, the
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
Solution:
a1 = 1, b1 = 1, c1 =
5
Hence, the
pair of linear equations are consistent.
Both
equations need three solutions to be expressed graphically.
From
equation (1),
x + y =5
x = 5 – y
|
x |
4 |
3 |
2 |
|
y |
1 |
2 |
3 |
From
equation (2),
x =
(10-2y)/2
|
x |
4 |
3 |
2 |
|
y |
1 |
2 |
3 |
(ii) x
- y =8, 3x - 3y = 16
Solution:
a1 = 1, b1 = -1, c1 =
8
Hence, the
pair of linear equations are inconsistent.
(iii) 2x
+ y – 6 = 0, 4x – 2y – 4 = 0
Solution:
a1 = 2, b1 = 1, c1 =
-6
Hence, the
pair of linear equations are consistent.
Both
equations need three solutions to be expressed graphically.
From
equation (1),
x = (6 –
y)/2
|
x |
0 |
1 |
2 |
|
y |
6 |
4 |
2 |
From
equation (2),
x = (4 +
2y)/4
|
x |
1 |
2 |
3 |
|
y |
0 |
2 |
4 |
(iv) 2x
- 2y - 2 = 0, 4x - 4y – 5 = 0
Solution:
a1 = 2, b1 = -2, c1 =
-2
a2 = 4, b2 = -4, c2 = -5
Hence, the
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
width of the rectangular garden = x m
Let the
length of the rectangular garden = y m
According
to Question,
Half of
the perimeter = 36 m
Perimeter/2
= 36
x + y =
36…………………(i)
The length
of the garden is 4
m more than the width,
so
y – x =
4…………………(ii)
Both
equations need three solutions to be expressed graphically.
From
equation (1),
x = 36 – y
|
x |
0 |
36 |
20 |
|
y |
36 |
0 |
16 |
From
equation (2),
y = x + 4
|
x |
0 |
-4 |
5 |
|
y |
4 |
0 |
9 |
The point
of intersection of the equations is (16, 20).
Hence, the length of the
garden is 20 m
and the 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
Solution:
The linear equation 2x + 3y – 8 = 0 intersects the linear equation x + 3y – 10 = 0 because,
a1 = 2, b1 = 3, c1 =
-8
a2 = 1, b2 = 3, c2 =
-10
(ii) parallel lines
Solution:
(iii) coincident lines
Solution:
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:
x – y + 1 = 0…………..(i)
3x + 2y – 12 = 0………..(ii)
Both equations need three solutions to be expressed
graphically.
From equation (1),
y = x + 1
|
x |
0 |
-1 |
2 |
|
y |
1 |
0 |
3 |
From equation (1),
3x + 2y – 12 = 0
|
x |
0 |
2 |
4 |
|
y |
6 |
3 |
0 |
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