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**NCERT Class 10 Mathematics Chapter 3 Pair of Linear Equations in two Variables**

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**Solutions****NCERT Class 10 Mathematics Chapter 3 Pair of Linear Equations in two Variables****Pair of Linear Equations in two Variables**

**Pair of Linear Equations in two Variables****Chapter – 3**

Exercise 3.1 |

**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.**

Ans: Let number of boys be x and the number of girls be y.

According to the given conditions

x + y = 10 and y = x + 4

We get the required pair of linear equations as

x + y-10 = 0, x – y + 4 = 0

Graphical solution

x + y – 10 = 0 … (1)

x | 2 | 5 |

Y = 10- x | 8 | 5 |

x – y + = …..(2)

x | 2 | 4 |

Y = x + 4 | 6 | 8 |

From the graph, we have x = 3, y = 7

Common solution of the two linear equations.

Hence, the number of boys = 3

and the number of girls = 7

**(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.**

Ans: Let the cost of one pencil be Rs. x and the cost of one pes be Rs. y

According to the given condition,

Cost of 5 pencils and 7 pens = Rs. 50

Cost of 7 pencils and 5 pens = Rs. 46

5x + 7y = 50 and 7x+5y=46

Hence, the required linear equations are

5x + 7y=50, 7x+5y = 46

Graphical solution 5 × + y = 50 …(1)

× | 2 | 3 | 6 |

Y= 50 – 5× /7 | 5.7 | 5 | 2.8 |

7× + 5y = 46 …(2)

× | 2 | 3 | 4 |

Y = 46 – 7 × /5 | 6.4 | 5 | 3.6 |

From the graph, we have the common solution x = 3, y = 5 Hence, the cost of one pencil = Rs. 3 and the cost of one pen = Rs.

**2. One compares the ratios a and find out whether the lines C2 representing the following pairs of linear equations intersect at a point, are parallel or coincident.**

**(i) 5x-4y+8=0… (1)**

**7x+6y-9=0… (2)**

Ans: 5x-4y+8=0

7x+6y-9=0

a₁ /b₂ 5/7, b₁/b₂ = -4/6 = -4/6 = -⅔

⇒ a₁/b₂ ≠ b₁/b₂

⇒ Lines represented by (1) and (2) intersect at a point

**(ii) 9x+3y+12=0 … (1)**

**18x+6y+ 24 = 0… (2)**

Ans: 9 × + 3y + 12 = 0

18 × + 6 y + 24 = 0

a₁/a₂ = 8/19 = ½ , b₁/b₂= 3/6 = ½ ⇒a₁a₂ /b₁/b₂ = C₁ /C ₂

⇒ Liner represented by (i) ans (ii) are coincident

**(iii) 6x-3y+10=0… (1)**

**2x-y+9=0… (2)**

Ans: 6x-3y+10=0

2x-y+9=0

a₁/a₂ = 6/2 = 3, b₁/b₂ = -3/-1 = 3 ,C₁ /C ₂ = 10/9

a₁/a₂ = b₁/b₂ ≠ C₁ /C

⇒ lines represented by (1) and (2) are parallel.

**3. On the comparing the rations a₁/a₂ = b₁/b₂ and C₁ /C find out whether the following pairs of lines questions are consistent for inconsistent.**

**(i) 3x+2y=5; 2x-3y=7**

Ans: 3× + 2y – 5 = 0

2 × – 3y – 7 = 0

a₁/a₂ = 3/2 ; b₁/b₂ = 2/-3 = -⅔ ⇒ a₁/a₂≠C₁ /C₂

⇒ The equations have a unique Hence, consistent

**(ii) 2x-3y = 8; 4x-6y=9**

Ans: 2× – 3y – 8 = 0

4× – 6y – 9 = 0

a₁/a₂ = 2/4 = ½ ; b₁b₂ = -3 /-6 = ½; C₁ /C₂

= -8/-9 = 8/9 ⇒ a₁/a₂ = b₁b₂ ⇒ C₁ /C₂

⇒ The equation have no solution.

Hence, inconsistent.

**(iii) 3/2 + ⅗y = 7; 9x-10y = 14**

Ans: 3/2 + ⅗y – 7 = 0

9x-10y – 14 = 0

a₁/a₂ = (3/2)/9 = ⅙ ; ≠ = (5/3)/-10 = ⅙ ⇒

a₁/a₂ ≠ b₁b₂

⇒ The equation have no solution. Hence, inconsistent.

**(iv) 5× – 3y = 11; – 10× + 6y = -22**

Ans: 5x – 3y – 11 ,= 0

10x – 6y – 22 = 0

→ Equations have infinitely many solutions. Hence, consistent.

**(v) 4/3× 2y = 8; +3y = 12**

Ans:

⇒ The equation have no solution. Hence, inconsistent.

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

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

Ans: x + y – 5 = 0 ……(i)

2x + ½ , b₁b₂ = ½, C₁ /C₂ = -5/-10 = ½

i.e a₁/a₂ = b₁b₂ = C₁ /C₂

Hence, the pair of linear equation is consistent (1) and (2) are same equations and hence the graph is a consistent straight Liner.

X | 1 | 3 |

Y = 5 – x | 4 | 2 |

Graph of liner equations (1) and (2) is the coincident line passing through the points (1,4) and (3,2) . every point on the line given a solution.

**(ii) X- y = 8, 3X- 3y = 16**

Ans: x-y -8= 0 (i)

3x- 3y -16 = 0…(ii)

a₁/a₂ = ⅓, b₁b₂= -1/-3, ⅓ C₁/C₂ = -8/-16,= ½ ; a₁/a₂ = b₁b₂ ≠ C₁/C₂

The pair of equations is inconsistent and the graph of two equations is a pair of parallel straight lines.

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

Ans: 2x + y – 6 = 0 (1)

4x – 2y – 4 = 0… (2)

a₁/a₂ = 2/4 = ½, b₁b₂ = 1/-2 = -½