Convert both sides to decimal for each option and look for the pair that agrees — but two quick observations remove most of the work first.

Observation 1 — the digits constrain the bases. Every digit must be less than its base. The left-hand number contains a 5, so \(x\gt5\); the right-hand number contains a 3, so \(y\gt3\). That alone eliminates option 1, where x = 5 and y = 3 are both illegal.

Observation 2 — test the remaining options.

\((1235)_{x}=x^{3}+2x^{2}+3x+5\)

\((3033)_{y}=3y^{3}+0y^{2}+3y+3\)

So x = 8 and y = 6, both sides representing the decimal number 669 — option 3.

Working option 3 out in full, since it is the answer:

\((1235)_{8}=1\times512+2\times64+3\times8+5\times1=512+128+24+5=669\)

\((3033)_{6}=3\times216+0\times36+3\times6+3\times1=648+0+18+3=669\)

A useful shortcut for problems of this kind. The two numbers have the same digit count, and the left one begins with 1 while the right begins with 3 — so the right-hand base must be appreciably smaller, since three of its highest power equals roughly one of the other's. Indeed \(3y^{3}\approx x^{3}\) gives \(x\approx1.44y\), and 8 against 6 fits that ratio far better than 7 against 5 or 9 against 7.

The underlying principle is positional notation: a digit's contribution is the digit multiplied by the base raised to its position index, counting from zero at the right. The same number has many representations, but its value is invariant — which is exactly what the question is testing.

Hence, x = 8 and y = 6.