## NCERT solutions for class 10 Maths

**Chapter 3**

**Pair of Linear Equations in Two Variables**

**Exercise 3.2**

**Ex 3.2 Question 1.**

**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 girls = x and the number of boys in the class = y respectively.

x + y = 10

x – y = 4

x + y = 10 x = 10 – y

Three solutions of this equation are

x | 5 | 4 | 6 |

y | 5 | 6 | 4 |

x – y = 4 x = 4 + y

Three solutions of this equation are

x | 5 | 4 | 3 |

y | 1 | 0 | -1 |

Two lines cross each other at the point (7, 3).

∴There are 7 girls and 3 boys in the class. So, x = 7 and y = 3.

(ii) Let the cost of one pencil = ₹ X and one pen = ₹ y respectively.

5x + 7y = 50

7x + 5y = 46

5x + 7y = 50 or x = (50-7y)/5

Three solutions of this equation are

x | 3 | 10 | -4 |

y | 5 | 0 | 10 |

7x + 5y = 46 or x = (46-5y)/7

Three solutions of this equation are

x | 8 | 3 | -2 |

y | -2 | 5 | 12 |

Two lines cross each other at the point (3, 5).

∴The cost of a pencil is 3/- and the cost of a pen is 5/- So, x = 3 and y = 5.

**Ex 3.2 Question 2.**

**On comparing the ratios a _{1}/a_{2},b_{1}/b_{1},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**

**Solutions:**

(i)5x−4y+8 = 0

7x+6y−9 = 0

Comparing these equations with a_{1}x + b_{1}y + c_{1} = 0 and 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, c_{2} = -9

(a_{1}/a_{2}) = 5/7

(b_{1}/b_{2}) = -4/6 = -2/3

(c_{1}/c_{2}) = 8/-9

Since, (a_{1}/a_{2}) ≠ (b_{1}/b_{2}) the given pair of equations intersect at exactly one point.

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

18x + 6y + 24 = 0

Comparing these equations with a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0, we ge

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

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

(a_{1}/a_{2}) = 9/18 = 1/2

(b_{1}/b_{2}) = 3/6 = 1/2

(c_{1}/c_{2}) = 12/24 = 1/2

Since (a_{1}/a_{2}) = (b_{1}/b_{2}) = (c_{1}/c_{2}) the given pair of equation are coincident.

(iii) 6x – 3y + 10 = 0

2x – y + 9 = 0

Comparing these equations with a_{1}x + b_{1}y + c_{1} = 0 and 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, c_{2} = 9

(a_{1}/a_{2}) = 6/2 = 3/1

(b_{1}/b_{2}) = -3/-1 = 3/1

(c_{1}/c_{2}) = 10/9

Since (a_{1}/a_{2}) = (b_{1}/b_{2}) ≠ (c_{1}/c_{2}) the given pair of equation are parallel to each other.

**Ex 3.2 Question 3.**

**On comparing the ratio, a _{1}/a_{2},b_{1}/b_{1},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**

**Solutions:**

(i) 3x + 2y = 5 and 2x – 3y = 7

Comparing these equations with a_{1}x + b_{1}y + c_{1} = 0 and 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, c_{2} = 7

(a_{1}/a_{2}) = 3/2

(b_{1}/b_{2}) =2/-3

(c_{1}/c_{2}) = -5/7

Since, (a_{1}/a_{2}) ≠ (b_{1}/b_{2}) the given pair of equations intersect each other at one point and they have only one possible solution. The equations are consistent.

(ii) 2x – 3y = 8 and 4x – 6y = 9

Comparing these equations with a_{1}x + b_{1}y + c_{1} = 0 and 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, c_{2} = -9

(a_{1}/a_{2}) = 2/4 = 1/2

(b_{1}/b_{2}) = -3/-6 = 1/2

(c_{1}/c_{2}) = -8/-9 = 8/9

Since (a_{1}/a_{2}) = (b_{1}/b_{2}) ≠ (c_{1}/c_{2}) the given pair of equations are parallel to each other and they have no possible solution. Hence, the equations are inconsistent.

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

Comparing these equations with a_{1}x + b_{1}y + c_{1} = 0 and 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, c_{2} = 14

(a_{1}/a_{2}) = 3/2

(b_{1}/b_{2}) =2/-3

(c_{1}/c_{2}) = -5/7

Since, (a_{1}/a_{2}) ≠ (b_{1}/b_{2}) the given pair of equations intersecting each other at one point and they have only one possible solution. Hence, the equations are consistent.

(iv) 5x – 3y = 11 and – 10x + 6y = –22

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

a_{2} = -10, b_{2} = 6, c_{2} = 22

(a_{1}/a_{2}) = 5/(-10) = -5/10 = -1/2

(b_{1}/b_{2}) = -3/6 = -1/2

(c_{1}/c_{2}) = -11/22 = -1/2

Since (a_{1}/a_{2}) = (b_{1}/b_{2}) = (c_{1}/c_{2}) the given pair of equations have infinite number of possible solutions. Hence, the equations are consistent.

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

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

a_{2} = 2, b_{2} = 3 , c_{2} = -12

(a_{1}/a_{2}) = 4/(3×2)= 4/6 = 2/3

(b_{1}/b_{2}) = 2/3

(c_{1}/c_{2}) = -8/-12 = 2/3

Since (a_{1}/a_{2}) = (b_{1}/b_{2}) = (c_{1}/c_{2}) the given pair of equations have infinite number of possible solutions. Hence, the equations are consistent.

**Ex 3.2 Question 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**

**Solutions:**

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

(a_{1}/a_{2}) = 1/2

(b_{1}/b_{2}) = 1/2

(c_{1}/c_{2}) = 1/2

Since (a_{1}/a_{2}) = (b_{1}/b_{2}) = (c_{1}/c_{2})the given pair of equations have infinite number of possible solutions. Hence, the equations are consistent.

For, x + y = 5 or x = 5 – y

For 2x + 2y = 10 or x = (10-2y)/2

The given pair of equations has infinite possible solutions.

(ii) x – y = 8 and 3x – 3y = 16

(a_{1}/a_{2}) = 1/3

(b_{1}/b_{2}) = -1/-3 = 1/3

(c_{1}/c_{2}) = 8/16 = 1/2

Since, (a_{1}/a_{2}) = (b_{1}/b_{2}) ≠ (c_{1}/c_{2}) the given pair of equations has no solution. Thus, the pair of linear equations is inconsistent.

(iii) 2x + y – 6 = 0 and 4x – 2y – 4 = 0

(a1/a2) = 2/4 = ½

(b1/b2) = 1/-2

(c1/c2) = -6/-4 = 3/2

Since (a_{1}/a_{2}) ≠ (b_{1}/b_{2}) the given pair of equations has only one solution. Hence, the pair of linear equations is consistent.

Now, for 2x + y – 6 = 0 or y = 6 – 2x

And for 4x – 2y – 4 = 0 or y = (4x-4)/2

Two lines are intersecting each other at only one point,(2,2). Hence, the solution of the given pair of equations is (2, 2).

(iv) x – 2y – 2 = 0 and 4x – 4y – 5 = 0

(a_{1}/a_{2}) = 2/4 = ½

(b_{1}/b_{2}) = -2/-4 = 1/2

(c_{1}/c_{2}) = 2/5

Since, (a_{1}/a_{2}) = (b_{1}/b_{2}) ≠ (c_{1}/c_{2}) the given pair of equations has no solution. Thus, the pair of linear equations is inconsistent.

**Ex 3.2 Question 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 garden is x and length is y.

y – x = 4

y + x = 36

y – x = 4 ⇒ y = x + 4

x | 0 | 8 | 12 |

y | 4 | 12 | 16 |

For y + x = 36 ⇒y = 36 – x

x | 0 | 36 | 16 |

y | 36 | 0 | 20 |

Two lines intersect each other at the point (16, 20). So, x = 16 and y = 20. Hence, the width of the garden is 16 m and the length is 20 m.

**Ex 3.2 Question 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**

**Solutions:**

**(i) Given the linear equation 2x + 3y – 8 = 0.**

To find another linear equation in two variables such that the geometrical representation of the pair so formed is intersecting lines, it should satisfy below condition;

(a_{1}/a_{2}) ≠ (b_{1}/b_{2})

So, the other linear equation can be 2x – 7y + 9 = 0,

(a_{1}/a_{2}) = 2/2 = 1

(b_{1}/b_{2}) = 3/-7

(ii) Parallel lines

(a_{1}/a_{2}) = (b_{1}/b_{2}) ≠ (c_{1}/c_{2})

So, the other linear equation can be 6x + 9y + 9 = 0,

(a_{1}/a_{2}) = 2/6 = 1/3

(b_{1}/b_{2}) = 3/9= 1/3

(c_{1}/c_{2}) = -8/9

(iii) Coincident lines

(a_{1}/a_{2}) = (b_{1}/b_{2}) = (c_{1}/c_{2})

So, the other linear equation can be 4x + 6y – 16 = 0,

(a_{1}/a_{2}) = 2/4 = 1/2 ,(b_{1}/b_{2}) = 3/6 = 1/2, (c_{1}/c_{2}) = -8/-16 = 1/2

**Ex 3.2 Question 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: Given, the equations for graphs are x – y + 1 = 0 and 3x + 2y – 12 = 0.**

**For, x – y + 1 = 0 or x = 1+y**

**For, 3x + 2y – 12 = 0 or x = (12-2y)/3**

The coordinates of the vertices of the triangle so formed are (2, 3), (-1, 0), and (4, 0).