Given
\[ \frac{49^{x+4}}{7^{y-5}} = 2401 \]
Express 𝑦 in terms of 𝑥, writing your answer in simplest form.
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SOLUTION:
We begin by writing out our equation:
\[ \frac{49^{x+4}}{7^{y-5}} = 2401 \]We can rewrite all the terms of our equation with the base 7:
\[ \frac{7^{2(x+4)}}{7^{y-5}} = 7^4 \]Since the numerator and denominator have the same base, we can simplify the fraction by keeping the base and subtracting the exponent in the denominator from the exponent in the numerator:
\[ 7^{\,2(x+4) - y + 5} = 7^4 \]Since both sides of the equation have the same base, we can set the exponents equal to each other.:
\[ 2(x+4) - y + 5 = 4 \]Rewriting this equation to have y in terms of x:
\[ y = 2(x+4) + 5 - 4 \]Simplifying this, we get our final answer:
\[\therefore \underline{\underline{y = 2x + 9 }}\]