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

The cost of $2\,kg$ of apples and $1\,kg$ of grapes on a day was found to be Rs. $160$. After a month,the cost of $4\,kg$ of apples and $2\,kg$ of grapes is Rs. $300$. Represent this situation as a pair of linear equations in two variables.

If the following pair of equations is consistent (unique solution),then find the value of $k$:
$x - ky = 2$
$3x + 2y = -5$

Solve the pair of linear equations $2x + y = 40$ and $x + y = 30$. If the solution satisfies the equation $y = mx + 6$,find the value of $m$.

$A$ fraction becomes $\frac{1}{4}$ if $2$ is subtracted from the numerator and $3$ is added to the denominator. If $6$ is added to the numerator and the denominator is multiplied by $3$,it becomes $\frac{2}{3}$. Find the fraction.

In $\Delta ABC$,$m \angle A = x$,$m \angle B = 3x$,and $m \angle C = y$. If $3y - 5x = 30$,show that the triangle is right-angled and find the measures of all three angles.