Solve the given equations and select the correct option.
$I.$ $7x + 6y = 110$
$II.$ $4x + 3y = 59$
$III.$ $x + z = 15$

Sachin and Rahul attempted to solve a quadratic equation. Sachin made a mistake in writing down the constant term and ended up with roots $(4, 3).$ Rahul made a mistake in writing down the coefficient of $x$ and got roots $(3, 2).$ The correct roots of the equation are:

Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{3} - 468 = 1729$
$II.$ $y^{2} - 1733 + 1564 = 0$

Solve the given two equations and select the correct option.
$I.$ $42x - 17y = -67$
$II.$ $7x - 12y = -26$

If $\alpha$ and $\beta$ are the roots of the equation $2x(2x + 1) = 1$,then $\beta$ is equal to

If ${x^2} + {y^2} = 25$ and $xy = 12$,then find the possible values of $x$.