The digit at the ten's place is $4$ in a two-digit number,and the product of both the digits is four times greater than the digit at the ten's place. Find the number.

Determine,algebraically,the vertices of the triangle formed by the lines $5x - y = 5$,$x + 2y = 1$,and $6x + y = 17$.

For which value$(s)$ of $\lambda$,do the pair of linear equations $\lambda x + y = \lambda^2$ and $x + \lambda y = 1$ have:
$(i)$ no solution?
$(ii)$ infinitely many solutions?
$(iii)$ a unique solution?

If a pair of equations $3x + ky - 9 = 0$ and $x + 2y - 3 = 0$ has infinite solutions,then $k = \ldots$

Devarshi got thrice the marks as obtained by Maharshi in the annual examination of mathematics of standard $10$. The sum of the marks obtained by them was $150$. Represent this situation as a pair of linear equations in two variables.

Solve the following pair of linear equations:
$\frac{x}{a} + \frac{y}{b} = a + b$
$\frac{x}{a^{2}} + \frac{y}{b^{2}} = 2, \quad a, b \neq 0$