System of exponential equations

CAT 2019 Slot 2 · Quantitative Ability · Medium · Exponents & Powers

This is a medium Quantitative Ability question from the CAT 2019 Slot 2 paper. It tests Exponents & Powers. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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If $5^x - 3^y = 13438$ and $5^{x-1} + 3^{y+1} = 9686$ , then $x+y$ equals

[TITA]

Answer

Explanation

Let $a = 5^x$ and $b = 3^y$ .

$a - b = 13438 \implies b = a - 13438$
$\frac{a}{5} + 3b = 9686 \implies a + 15b = 48430$

Substitute $b$ into the second equation: $a + 15(a - 13438) = 48430$ $16a - 201570 = 48430$ $16a = 250000 \implies a = 15625 = 5^6 \implies x = 6$ $b = 15625 - 13438 = 2187 = 3^7 \implies y = 7$

Thus $x + y = 6 + 7 = 13$ .

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