Find the equation of the straight line which passes through the point $\left(2, -3\right)$ and cuts off intercepts on the axes equal in magnitude and of the same sign.
Let
intercept made by the required line on the $X$ axis $= a$
intercept made by the required line on the $Y$ axis $= b$
the equation of the required line be of the form: $\;$ $\dfrac{x}{a} + \dfrac{y}{b} = 1$
Given: $\;$ $a = b$
$\therefore \;$ The equation of the required line becomes $\;$ $\dfrac{x}{a} + \dfrac{y}{a} = 1$
i.e. $\;$ $x + y = a$ $\;\;\; \cdots \; (1)$
The required equation passes through the point $\left(2, -3\right)$.
Substituting the point in equation $(1)$ gives
$2 - 3 = a$ $\implies$ $a = -1$
Substituting the value of $a$ in equation $(1)$ gives
$x + y = -1$
$\therefore \;$ The equation of the required line is $\;$ $x + y + 1 = 0$