Question

2/x + 12/y = 1/2

4/x +12/y = 13/20​

Answer 1

Given

The Given linear equations are :

$\\ \bullet \bf \: \frac{2}{x} + \frac{12}{y} = \frac{1}{2} \: \: ...(1)$

$\\ \bullet \bf \: \frac{4}{x} + \frac{12}{y} = \frac{13}{20} \: \: ...(2)$

To Find :

• The value of x and y =?

Solution :

$\\ \bullet \bf \: \frac{2}{x} + \frac{12}{y} = \frac{1}{2} \: \: ...(1)$

$\\ \star\pink{\underline{\sf{Getting \: reciprocal \: of \: equation \: (1) ,\: we \: get \:}}}$

$\\ \implies \sf \: \frac{x}{2} + \frac{y}{12} = 2$

$\\ \star\pink{\underline{\sf{On \: cross \: multiplication \: of \: equation (1) \: ,we \: get }}}$

$\\ \implies \sf \: \frac{x \times 6}{2 \times 6} + \frac{y}{12} = 2 \\ \\ \implies \sf \: \frac{6x + y}{12} = 2 \\ \\ \implies \sf \: 6x + y = 24 \\ \\ \: \: \: \: \: \: \implies \boxed{ \sf{6x + y = 24}} \: \: ...(3)$

$\\ \bullet \bf \: \frac{4}{x} + \frac{12}{y} = \frac{13}{20}$

$\\ \star\pink{\underline{\sf{Getting \: reciprocal \: of \: equation (2) \: , we \: get }}}$

$\\ \implies \sf \: \frac{x}{4} + \frac{y}{12} = \frac{20}{13}$

$\\ \star\pink{\underline{\sf{On \: cross \: multiplication \: of \: equation (2) , \: we \: get }}}$

$\\ \implies \sf \: \frac{x \times 3}{4 \times 3} + \frac{y}{12} = \frac{20}{13} \\ \\ \implies \sf \: \frac{3x + y}{12} = \frac{20}{13} \\ \\ \implies \sf \: 13(3x + y) = 20 \times 12 \\ \\ \implies \sf \: 39x + 13y = 240 \\ \\ \implies \boxed{ \sf{39x + 13y = 240}} \: ....(4)$

$\\ \star\pink{\underline{\sf{Multiplying \: in \: equation (3) \: by \: 13 , \: we \: get}}}$

$\implies \boxed{ \sf{\:78 x + 13y = 312}} \: \: ...(5)$

$\\ \star\pink{\underline{\sf{Subtracting \: equations (5) \: and \: (4)}}}$

$\implies \sf78x + \cancel{13y }= 312 \: \: ..(5)$

$\ \: \: \: \: \: \: \: \: \: \: \: \sf \: 39x + \cancel{13y} = 240...(4)$

____________________________

$\implies \sf \: 39x = 72 \\ \\ \implies \sf \: x = \frac{72}{39} \\ \\ \implies \sf \: x = \frac{24}{13} \\ \\ \implies \boxed{ \sf{x = \frac{24}{13} }}$

$\\ \star\pink{\underline{\sf{Substitute \: the \: value \: x \: in \: equation (1), we \: get}}}$

$\\ \implies \sf \: 6x + y = 24 \\ \\ \implies \sf \: 6 \bigg( \frac{24}{13} \bigg) + y = 24 \\ \\ \implies \sf \: 6 \times 24 + 13y = 24 \times 13 \\ \\ \implies \sf \: 13y = 24 \times 13 - 6 \times 24 \\ \\ \implies \sf \: 13y = 24(13 - 6) \\ \\ \implies \sf \: 13y = 24 \times 7 \\ \\ \implies \sf \: 13y = 168 \\ \\ \implies \sf \: y = \frac{168}{13} \\ \\ \implies \boxed{ \sf{ y = \frac{168}{13} }}$

The value of x and y is :

$\red \bigstar \boxed{ \sf{x = \frac{24}{13} \: \: and \: \: y = \frac{168}{13} }}$