Question

solve for x and y .​

Answer 1

ANSWER.

=> x = ½ and y = 5/4

EXPLANATION.

$\sf \to \: \dfrac{1}{2(x + 2y)} + \dfrac{5}{3(3x - 2y)} = \dfrac{ - 3}{2} \\ \\ \sf \to \: \dfrac{5}{4(x + 2y)} - \dfrac{3}{5(3x - 2y)} = \dfrac{61}{60}$

$\sf \to \: let \: we \: assume \: that \\ \\ \sf \to \: \frac{1}{ {(x + 2y)}} = a \: \: \: \: and \: \: \: \dfrac{1}{(3x - 2y)} = b$

$\sf \to \: \dfrac{a}{2} + \dfrac{5b}{3} = \dfrac{ - 3}{2} \\ \\ \sf \to \: \frac{3a + 10b}{ \cancel{6}} = \frac{ - 3}{ \cancel{2}} \\ \\ \sf \to \: 3a + 10b \: = - 9 \: ......(1)$

$\sf \to \: \dfrac{5a}{4} - \dfrac{3b}{5} = \dfrac{61}{60} \\ \\ \sf \to \: \dfrac{25a - 12b}{ \cancel{20}} = \frac{61}{ \cancel{60}} \\ \\ \sf \to \: 75a - 36b \: = 61 \: .......(2)$

$\sf \to \: from \: equation \: (1) \: and \: (2) \: we \: get$

$\sf \to \: multiply \: equation \: (1) \: by \: 36 \\ \\ \sf \to \: multiply \: equation \: (2) \: by \: 10$

$\sf \to \: we \: get \\ \\ \sf \to \: 108a + 360b \: = - 324....(1) \\ \\ \sf \to \: 750a - 360b \: = 610.......(2) \\ \\ \sf \to \: we \: get \\ \\ \sf \to \: 858a = 286 \\ \\ \sf \to \: a \: = \dfrac{1}{3} \\ \\ \sf \to \: put \: the \: value \: of \: a \: = \dfrac{1}{3} \: in \: equation \: (1) \\ \\ \sf \to \: we \: get \\ \\ \sf \to \: \cancel{3} \times \dfrac{1}{ \cancel{3}} + 10b = - 9 \\ \\ \sf \to 10b \: = - 10 \\ \\ \sf \to \: b \: = - 1$

$\sf \to \: as \: we \: can \: assume \: that \\ \\ \sf \to \: \dfrac{1}{x + 2y} = a \implies \: \dfrac{1}{x + 2y} = \dfrac{1}{3} \\ \\ \sf \to \: \dfrac{1}{3x - 2y} = \: b \implies \: \dfrac{1}{3x - 2y} = \dfrac{ - 1}{1}$

$\sf \to \: x \: + 2y = 3 \: .....(3) \\ \\ \\ \sf \to \: - 3x + 2y = 1 \: .....(4) \\ \\ \sf \to \: we \: get \: \\ \\ \sf \to \: multiply \: equation \: (3) \: by \: 3 \\ \\ \sf \to \: multiply \: equation \: (4) \: by \: 1 \\ \\ \sf \to \: + 3x + 6y = 9 \\ \\ \sf \to \: - 3x + 2y = 1 \\ \\ \sf \to \: 8y = 10 \implies \: y \: = \dfrac{5}{4} \\ \\ \sf \to \: put \: y \: = \frac{5}{4} in \: equation \: (3) \: we \: get \: \\ \\ \sf \to \: x + 2 \times \frac{5}{4} = 3 \\ \\ \sf \to \: x + \dfrac{5}{2} = 3 \\ \\ \sf \to \: x \: = 3 - \dfrac{5}{2} \\ \\ \sf \to \: x \: = \dfrac{1}{2} \\ \\ \sf \to \: \green{{ \underline{value \: of \: x \: = \dfrac{1}{2} \: and \: y \: = \dfrac{5}{4} }}}$