Question

Solve for $x$ and $y$: $47x+31y=63,31x+47y=15$

Answer 1

Given $47x+31y=63$ ---(i)
$31x+47y=15$ ---(ii)

Here, we observe that the coefficients of $x$ and $y$ in equation(i) are interchanged as $y$ and $x$ coefficients in equation(ii).

To make the $x$ coefficients equal, apply the following operation:

Multiplying (i) by $31$ and multiplying (ii) by $47$, we get

$1457x+961y=1953$ ---(iii)

$1457x+2209y=705$ ---(iv)

Subtracting (iv) from (iii),we get
$1457x+961y-1457x-2209y=1953-705$

$-1248y=1248$

$\Rightarrow y=-1$

Putting $y=-1$ in (i), we get

$47x+31\times -1=63$
$47x=63+31$

$\Rightarrow 47x=94$

$\Rightarrow x=2$

Therefore $x=2$ and $y=-1$