# 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__

__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

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 ****a _{1}/a_{2}, b_{1}/b_{2}**

**and**

**c**

_{1}/c_{2}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****:**

a_{1} = 5, b_{1} = -4, c_{1} =
8

_{2}= 7, b

_{2}= 6, c

_{2}= -9

Hence, the lines represented by the pair of equations
intersect at a point.

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

**18x + 6y +24 = 0**

**Solution****:**

a_{1} = 9, b_{1} = 3, c_{1} =
12

a_{2} = 18, b_{2} = 6, c_{2} =
24

Hence pair of linear equations are coincident.

**(****iii****)****6x - 3y + 10 = 0**

**2x - y + 9 = 0**

**Solution****:**

a_{1} = 6, b_{1} = -3, c_{1} =
10

_{2}= 2, b

_{2}= -1, c

_{2}= 9

Hence,
pair of linear equations are parallel.

**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**

**Solution****:**

a_{1} = 3, b_{1} = 2, c_{1} =
5

_{2}= 2, b

_{2}= -3, c

_{2}= 7

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****:**

a_{1} = 2, b_{1} = -3, c_{1} =
8

_{2}= 4, b

_{2}= -6, c

_{2}= 9

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

a_{1} = 9, b_{1} = 10, c_{1} =
42

_{2}= 9, b

_{2}= -10, c

_{2}= 14

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****:**

a_{1} = 5, b_{1} = -3, c_{1} =
11

_{2}= -10, b

_{2}= 6, c

_{2}= -22

Hence, the
pair of linear equations are consistent.

**(****v****)**** (4/3)x + 2y = 8, 2x + 3y = 12**

**Solution****:**

a_{1} = 4/3, b_{1} = 2, c_{1}
= 8

_{2}= 2, b

_{2}= 3, c

_{2}= 12

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****:**

a_{1} = 1, b_{1} = 1, c_{1} =
5

_{2}= 2, b

_{2}= 2, c

_{2}= 10

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 |

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****:**

a_{1} = 2, b_{1} = -2, c_{1} =
-2

a_{2} = 4, b_{2} = -4, c_{2} =
-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,

a_{1} = 2, b_{1} = 3, c_{1} =
-8

a_{2} = 1, b_{2} = 3, c_{2} =
-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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