Solve the following pairs of equations:
$ 43 x+67 y=-24 $
$ 67 x+43 y=24 $

Given:

\( 43 x+67 y=-24 \)

\( 67 x+43 y=24 \)

To do:

We have to solve the given pairs of equations.

Solution:

\( 43 x+67 y=-24 \)......(i)

\( 67 x+43 y=24 \).........(ii)

Multiplying (i) by 43 and (ii) by 67 and subtracting the results, we get,

$43(43x+67y)=43(-24)$

$43^2x+43(67)y=24(-43)$.........(iii)

$67(67x+43y)=67(24)$

$67^2x+43(67)y=24(67)$.......(iv)

Subtracting (iii) from (iv), we get,

$(67^2-43^2)x=24(67+43)$

$(67+43)(67-43)x=24(110)$

$110(24)x=24(110)$

$x=1$

This implies,

$43(1)+67y=-24$

$67y=-24-43$

$67y=-67$

$y=-1$

Therefore,

$x=1$

$y=-1$