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Exercise - 3.2

Pair of Linear Equations in two Variables

**Question-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.

(i) Let the no. of girls be x and no. of boys be y. The algebraic representation is given by the following equations: x + y = 10 ......(i) x – y = 4 ......(ii) Lets represent these equations in graphically: x + y = 10 .......(i)

x | 0 | 10 |

y | 10 | 0 |

x | 0 | 4 |

y | -4 | 0 |

(ii) Let the cost of pencil be x and cost of pen be y. The algebraic representation is given by the following equations: 5x + 7y = 50 ......(i) 7x + 4y = 46 ......(ii) Lets represent these equations in graphically: 5x + 7y = 50 ......(i)

x | 3 | 10 |

y | 5 | 0 |

x | 3 | 8 |

y | 5 | -2 |

**Question-2 :-** On comparing the ratios a₁/a₂, b₁/b₂ and c₁/c₂ find out whether 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

(i) 5x – 4y + 8 = 0; 7x + 6y – 9 = 0 a₁ = 5, b₁ = -4, c₁ = 8 a₂ = 7, b₂ = 6, c₂ = -9 Here, a₁/a₂ = 5/7, Also, b₁/b₂ = -4/6 and c₁/c₂ = 8/(-9) Therefore, a₁/a₂ ≠ b₁/b₂ ≠ c₁/c₂ Now, these linear equations are intersecting each other at one point and its have only one solution.

(ii) 9x + 3y + 12 = 0; 18x + 6y + 24 = 0 a₁ = 9, b₁ = 3, c₁ = 12 a₂ = 18, b₂ = 6, c₂ = 24 Here, a₁/a₂ = 9/18 = 1/2, Also, b₁/b₂ = 3/6 = 1/2 and c₁/c₂ = 12/24 = 1/2 Therefore, a₁/a₂ = b₁/b₂ = c₁/c₂ Now, these linear equations are coincident and its have many solution.

(iii) 6x – 3y + 10 = 0; 2x – y + 9 = 0 a₁ = 6, b₁ = -3, c₁ = 10 a₂ = 2, b₂ = -1, c₂ = 9 Here, a₁/a₂ = 6/2 = 3, Also, b₁/b₂ = -3/(-1) = 3 and c₁/c₂ = 10/9 Therefore, a₁/a₂ = b₁/b₂ ≠ c₁/c₂ Now, these linear equations are parallel each other and its have no solution.

**Question-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

(ii) 2x – 3y = 8 ; 4x – 6y = 9

(iii) 3x/2 + 5y/3 = 7; 9x - 10y = 14

(iv) 5x – 3y = 11 ; – 10x + 6y = –22

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

(i) 3x + 2y = 5 ; 2x – 3y = 7 3x + 2y - 5 = 0 ; 2x – 3y - 7 = 0 a₁ = 3, b₁ = 2, c₁ = -5 a₂ = 2, b₂ = -3, c₂ = -7 Here, a₁/a₂ = 3/2, Also, b₁/b₂ = 2/(-3) and c₁/c₂ = -5/(-7)= 5/7 Therefore, a₁/a₂ ≠ b₁/b₂ ≠ c₁/c₂ Now, these linear equations are intersecting each other at one point and its have only one solution. So, It is consitent.

(ii) 2x – 3y = 8 ; 4x – 6y = 9 2x – 3y - 8 = 0 ; 4x – 6y - 9 = 0 a₁ = 2, b₁ = -3, c₁ = -8 a₂ = 4, b₂ = -6, c₂ = -9 Here, a₁/a₂ = 2/4 = 1/2, Also, b₁/b₂ = -3/(-6) = 1/2 and c₁/c₂ = -8/(-9) = 8/9 Therefore, a₁/a₂ = b₁/b₂ ≠ c₁/c₂ Now, these linear equations are parallel to each other and its have only no possible solution. So, It is inconsitent.

(iii) 3x/2 + 5y/3 = 7; 9x - 10y = 14 3x/2 + 5y/3 - 7 = 0; 9x - 10y - 14 = 0 a₁ = 3/2, b₁ = 5/3, c₁ = -7 a₂ = 9, b₂ = -10, c₂ = -14 Here, a₁/a₂ = (3/2)/9 = 1/6, Also, b₁/b₂ = (5/3)/(-10) = 1/(-6) and c₁/c₂ = -7/(-14) = 1/2 Therefore, a₁/a₂ ≠ b₁/b₂ ≠ c₁/c₂ Now, these linear equations are intersecting each other at one point and its have only one solution. So, It is consitent.

(iv) 5x – 3y = 11 ; – 10x + 6y = –22 5x – 3y - 11 = 0 ; – 10x + 6y + 22 = 0 a₁ = 5, b₁ = -3, c₁ = -11 a₂ = -10, b₂ = 6, c₂ = 22 Here, a₁/a₂ = 5/(-10) = 1/(-2), Also, b₁/b₂ = (-3)/6 = 1/(-2) and c₁/c₂ = -11/22 = 1/(-2) Therefore, a₁/a₂ = b₁/b₂ = c₁/c₂ Now, these linear equations are coincident and its have many solution. So, It is consitent.

(v) 4x/3 + 2y = 8; 2x + 3y = 12 4x/3 + 2y - 8 = 0; 2x + 3y - 12 = 0 a₁ = 4/3, b₁ = 2, c₁ = -8 a₂ = 2, b₂ = 3, c₂ = -12 Here, a₁/a₂ = (4/3)/2 = 2/3, Also, b₁/b₂ = 2/3 and c₁/c₂ = -8/(-12) = 2/3 Therefore, a₁/a₂ = b₁/b₂ = c₁/c₂ Now, these linear equations are coincident and its have many solution. So, It is consitent.

**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

(i) x + y = 5, 2x + 2y = 10 x + y - 5 = 0 ; 2x + 2y - 10 = 0 a₁ = 1, b₁ = 1, c₁ = -5 a₂ = 2, b₂ = 2, c₂ = -10 Here, a₁/a₂ = 1/2, Also, b₁/b₂ = 1/2 and c₁/c₂ = -5/(-10)= 1/2 Therefore, a₁/a₂ = b₁/b₂ = c₁/c₂ Now, these linear equations are coincident and its have many solution. So, It is consitent. Lets represent these equations in graphically: x + y = 5 ...........(i)

x | 0 | 5 |

y | 5 | 0 |

x | 0 | 5 |

y | 5 | 0 |

(ii) x – y = 8, 3x – 3y = 16 x – y - 8 = 0 ; 3x – 3y - 16 = 0 a₁ = 1, b₁ = -1, c₁ = -8 a₂ = 3, b₂ = -3, c₂ = -16 Here, a₁/a₂ = 1/3, Also, b₁/b₂ = -1/(-3) = 1/3 and c₁/c₂ = -8/(-16) = 1/2 Therefore, a₁/a₂ = b₁/b₂ ≠ c₁/c₂ Now, these linear equations are parallel to each other and its have only no possible solution. So, It is inconsitent.

(iii) 2x + y – 6 = 0, 4x – 2y – 4 = 0 a₁ = 2, b₁ = 1, c₁ = -6 a₂ = 4, b₂ = -2, c₂ = -4 Here, a₁/a₂ = 2/4 = 1/2, Also, b₁/b₂ = 1/(-2) and c₁/c₂ = -6/(-4) = 3/2 Therefore, a₁/a₂ ≠ b₁/b₂ ≠ c₁/c₂ Now, these linear equations are intersecting each other at one point and its have only one solution. So, It is consitent. Lets represent these equations in graphically: 2x + y = 6 ...........(i)

x | 0 | 3 |

y | 6 | 0 |

x | 0 | 1 |

y | -2 | 0 |

(iv) 2x – 2y – 2 = 0, 4x – 4y – 5 = 0 a₁ = 2, b₁ = -2, c₁ = -2 a₂ = 4, b₂ = -4, c₂ = -5 Here, a₁/a₂ = 2/4 = 1/2, Also, b₁/b₂ = (-2)/(-4) = 1/2 and c₁/c₂ = -2/(-5) = 2/5 Therefore, a₁/a₂ = b₁/b₂ ≠ c₁/c₂ Now, these linear equations are parallel to each other and its have only no possible solution. So, It is inconsitent.

**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.

Let the width of garden and length of garden be araxand y respectively. According to question linear equations are: y = x + 4 x - y = -4 ........(i) y + x = 36 x + y = 36 ........(ii) Lets represent these equations in graphically: x - y = -4 ...........(i)

x | 0 | -4 |

y | 4 | 0 |

x | 0 | 36 |

y | 36 | 0 |

**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

(i) For intersecting lines, a₁/a₂ ≠ b₁/b₂ ≠ c₁/c₂ Given : 2x + 3y - 8 = 0 ....(i) According to condition the second line is : 3x + 5y - 15 = 0 ....(ii) a₁ = 2, b₁ = 3, c₁ = -8 a₂ = 3, b₂ = 5, c₂ = -15 Here, a₁/a₂ = 2/3, Also, b₁/b₂ = 3/5 and c₁/c₂ = -8/(-15) = 8/15 Hence, a₁/a₂ ≠ b₁/b₂ ≠ c₁/c₂.

(ii) For parallel lines, a₁/a₂ = b₁/b₂ ≠ c₁/c₂ Given : 2x + 3y - 8 = 0 ....(i) According to condition the second line is : 4x + 6y - 18 = 0 ....(ii) a₁ = 2, b₁ = 3, c₁ = -8 a₂ = 4, b₂ = 6, c₂ = -18 Here, a₁/a₂ = 2/4, Also, b₁/b₂ = 3/6 and c₁/c₂ = -8/(-18) = 4/9 Hence, a₁/a₂ = b₁/b₂ ≠ c₁/c₂.

(iii) For coincident lines, a₁/a₂ = b₁/b₂ = c₁/c₂ Given : 2x + 3y - 8 = 0 ....(i) According to condition the second line is : 6x + 9y - 24 = 0 ....(ii) a₁ = 2, b₁ = 3, c₁ = -8 a₂ = 6, b₂ = 9, c₂ = -24 Here, a₁/a₂ = 2/6 = 1/3, Also, b₁/b₂ = 3/9 = 1/3 and c₁/c₂ = -8/(-24) = 1/3 Hence, a₁/a₂ = b₁/b₂ = c₁/c₂.

**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.

x – y + 1 = 0 x - y = -1 ........(i) 3x + 2y – 12 = 0 3x + 2y = 12 ........(ii) Lets represent these equations in graphically: x - y = -1 ........(i)

x | 0 | -1 |

y | 1 | 0 |

x | 0 | 4 |

y | 6 | 0 |

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