Question

5.Solve for x and y: 83x + 85y = 337, 85x + 83y = 335​

Answer 1

$\large\underline{\sf{Solution-}}$

Given pair of equations are

$\rm :\longmapsto\:83x + 85y = 337 - - - (1)$

and

$\rm :\longmapsto\:85x + 83y = 335 - - - (2)$

On adding equation (1) and (2), we get

$\rm :\longmapsto\:168x + 168y = 672$

$\rm :\longmapsto\:168(x +y) = 672$

$\bf\implies \:x + y = 4 - - - - (3)$

On Subtracting equation (2) from equation (1), we get

$\rm :\longmapsto\: - 2x + 2y = 2$

$\rm :\longmapsto\: 2(- x + y) = 2$

$\bf :\longmapsto\: - x + y = 1 - - - - (4)$

On adding equation (3) and (4), we get

$\rm :\longmapsto\:2y = 5$

$\bf\implies \:y = \dfrac{5}{2}$

On substituting the value of y, we get

$\rm :\longmapsto\:x + \dfrac{5}{2} = 4$

$\rm :\longmapsto\:x = 4 - \dfrac{5}{2}$

$\rm :\longmapsto\:x = \dfrac{8 - 5}{2}$

$\bf\implies \:x = \dfrac{3}{2}$

Verification :-

Consider equation (1),

$\rm :\longmapsto\:83x + 85y = 337$

On substituting the values of x and y, we have

$\rm :\longmapsto\:83 \times \dfrac{3}{2} + 85 \times \dfrac{5}{2} = 337$

$\rm :\longmapsto\: \dfrac{249}{2} + \dfrac{425}{2} = 337$

$\rm :\longmapsto\: \dfrac{249 + 425}{2} = 337$

$\rm :\longmapsto\: \dfrac{674}{2} = 337$

$\rm :\longmapsto\:337 = 337$

Hence, Verified

Consider Equation (2),

$\rm :\longmapsto\:85x + 83y = 335$

On substituting the values of x and y, we get

$\rm :\longmapsto\:85 \times \dfrac{3}{2} + 83 \times \dfrac{5}{2} = 335$

$\rm :\longmapsto\: \dfrac{255}{2} + \dfrac{415}{2} = 335$

$\rm :\longmapsto\: \dfrac{255 + 415}{2} = 335$

$\rm :\longmapsto\: \dfrac{670}{2} = 335$

$\rm :\longmapsto\:335 = 335$

Hence, Verified