Question 1: Form a pair of linear equations in the following problems, and find their solutions graphically.

(a) 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.

**Solution:** Let us assume that number of boys = x and number of girls = y. We get following equations as per question:

`x + y = 10`

Or, `y = 10 – x` ………(1)

This equation will give following values for x and y;

x | 1 | 2 | 3 | 4 |

y | 9 | 8 | 7 | 6 |

`y = x + 4` ………..(2)

This equation will give following values for x and y;

x | 1 | 2 | 3 | 4 |

y | 5 | 6 | 7 | 8 |

Following graph is plotted for the given pair of linear equations:

Number of boys = 3 and number of girls = 7

(b) 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.

**Solution:** Let us assume that price of a pencil is x and that of a pen is y. We get following equations as per question:

`5x + 7y = 50` ………(1)

This equation will give following values for x and y;

x | 1 | 2 | 3 | 4 |

y | 6.4 | 5.7 | 5 | 4.2 |

`7x + 5y = 46` ……….(2)

This equation will give following values for x and y;

x | 1 | 2 | 3 | 4 |

y | 7.8 | 6.4 | 5 | 3.6 |

Following graph is plotted for the given pair of linear equations.

Price of one pencil = Rs. 3 and Price of one pen = Rs. 5

Question 2: On comparing the ratios `(a_1)/(a_2)`, `(b_1)/(b_2)` and `(c_1)/(c_2)` find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident.

(a) `5x – 4y + 8 = 0` and `7x + 6y – 9 = 0`

**Solution:** In the given pair of linear equations;

`(a_1)/(a_2)=5/7`

`(b_1)/(b_2)=-4/6=-2/3`

It is clear that;

`(a_1)/(a_2)≠(b_1)/(b_2)`

Hence the lines representing the given pair of linear equations intersect at a point.

(b) `9x + 3y + 12 = 0` and `18x + 6y + 24 = 0`

**Solution:** In the given pair of linear equations;

`(a_1)/(a_2)=(9)/(18)=1/2`

`(b_1)/(b_2)=3/6=1/2`

`(c_1)/(c_2)=(12)/(24)=1/2`

It is clear that;

`(a_1)/(a_2)=(b_1)/(b_2)=(c_1)/(c_2)`

Hence the lines representing the given pair of linear equations will be coincident.

(c) `6x – 3y + 10 = 0` and `2x- y + 9 = 0`

**Solution:** For the given pair of linear equations;

`(a_1)/(a_2)=6/2=3`

`(b_1)/(b_2)=(-3)/(-1)=3`

`(c_1)/(c_2)=(10)/(9)`

It is clear that;

`(a_1)/(a_2)=(b_1)/(b_2)≠(c_1)/(c_2)`

Hence the lines representing the given pair of linear equations will be parallel.

Question 3: On comparing the ratios `(a_1)/(a_2)`, `(b_1)/(b_2)` and `(c_1)/(c_2)` find out whether the following pairs of linear equations are consistent or inconsistent.

(a) `3x + 2y = 5` and `2x – 3y = 7`

**Solution:** For the given pair of linear equations;

`(a_1)/(a_2)=3/2`

`(b_1)/(b_2)=(2)/(-3)`

It is clear that;

`(a_1)/(a_2)≠(b_1)/(b_2)`

Hence, the given pair of linear equations is consistent.

(b) `2x – 3y = 8` and `4x – 6y = 9`

**Solution:** For the given pair of linear equations;

`(a_1)/(a_2)=2/4=1/2`

`(b_1)/(b_2)=(-3)/(-6)=1/2`

`(c_1)/(c_2)=8/9`

It is clear that;

`(a_1)/(a_2)=(b_1)/(b_2)≠(c_1)/(c_2)`

Hence the given pair of linear equations is inconsistent.

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

**Solution:** For the given pair of linear equations;

`(a_1)/(a_2)=(3)/(2)÷9=(3)/(18)`

`(b_1)/(b_2)=(5)/(3)÷-10=-5/6`

It is clear that;

`(a_1)/(a_2)≠(b_1)/(b_2)`

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

**Solution:** For the given pair of linear equations;

`(a_1)/(a_2)=(4)/(3)÷2=(2)/(3)`

`(b_1)/(b_2)=2/3`

`(c_1)/(c_2)=2/3`

It is clear that;

`(a_1)/(a_2)=(b_1)/(b_2)= (c_1)/(c_2)`

Hence the given pair of linear equations is dependent and consistent.

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