Solve the following mathematical problems:

$s o l v e : \frac{x}{2} + \frac{y}{3} = 1, \frac{x}{3} + \frac{y}{2} = 1$

Created: 5 months ago | Updated: 1 month ago

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$G i v e n, \frac{x}{2} + \frac{y}{3} = 1 . . . . . . . . (i) & \frac{x}{3} + \frac{y}{2} = 1 . . . . . . . . . . . . . (i i) F r o m e q u a t i o n (i) \frac{x}{2} + \frac{y}{3} = 1 \Rightarrow \frac{3 x + 2 y}{6} = 1 ∴ 3 x + 2 y = 6 . . . . . . . . . . . . (i i i) F r o m e q u a t i o n (i i) \frac{x}{3} + \frac{y}{2} = 1 \Rightarrow \frac{2 x + 3 y}{6} = 1 ∴ 2 x + 3 y = 6 . . . . . . . . . . . . . . . (i v) N o w, m u l t i p l y i n g e q u a t i o n (i i i) b y 3 a n d e q u a t i o n (i v) b y 2 a n d s u b t r a c t i n g e q u a t i o n (i v) f r o m (i i i), w e g e t 9 x + 6 y = 18 - 4 x + 6 y = 12 5 x = 6$

$∴ x = \frac{5}{6}$

Putting value of x in equation (iii) we get

$3 x + 2 y = 6 \Rightarrow 3 \times \frac{6}{5} + 2 y = 6 \Rightarrow \frac{18}{5} + 2 y = 6 \Rightarrow 2 y = 6 - \frac{18}{5} \Rightarrow 2 y = \frac{30 - 18}{5} = \frac{12}{5} ∴ y = \frac{6}{5} ∴ x = \frac{6}{5} & y = \frac{6}{5}$

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simplify: $\frac{5 x + 2}{x^{2} - x - 20} + \frac{2 x - 1}{x^{2} - 4 x - 5}$

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Solve the following mathematical problems:

In a survey at an airport, 55 travelers said that last year they had been to Spain, 53 to Fr and 79 to Germany, 18 had been to Spain and France, 17 to Spain and Germany and 2 France and Germany, while 10 had to all three countries. How many travelers took part survey?

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Solve the following mathematical problems:

A shopkeeper sells two shirts at the same price. He makes 10% profit on one and loses 10 the other. How much in percentage does he gain or lose?

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Solve the following mathematical problems:

The length of a tangent from a point A at distance 5cm from the center of the circle is 4cm. Find the radius of the circle

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Solve the following mathematical problems:

Find the H.C.F. of $x^{3} - 16 x, 2 x^{2} + 9 x^{2} + 4 x, 2 x^{3} + x^{3} - 28 x$

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Solve the following mathematical problems:

solve: x2+y3=1, x3+y2=1

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$s o l v e : \frac{x}{2} + \frac{y}{3} = 1, \frac{x}{3} + \frac{y}{2} = 1$

simplify: $\frac{5 x + 2}{x^{2} - x - 20} + \frac{2 x - 1}{x^{2} - 4 x - 5}$

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