3 x 10^3 + 2 x 10^1 + 4 x 10^x + 5 x 10^y = 3024.05, where x and y are integers. Write down the value of x and y.

Steps to Solve

1. Convert the known values

   Break down the constants in the equation:

   • $3 \times 10^3 = 3000$
   • $2 \times 10^1 = 20$

   The equation becomes: $$ 3000 + 20 + 4 \times 10^x + 5 \times 10^y = 3024.05 $$

2. Combine the known values

   Calculate the sum of the known values:

   $$ 3000 + 20 = 3020 $$

   Thus, the equation simplifies to: $$ 3020 + 4 \times 10^x + 5 \times 10^y = 3024.05 $$

3. Set up for the unknowns

   Rearranging the equation to isolate the unknowns:

   $$ 4 \times 10^x + 5 \times 10^y = 3024.05 - 3020 $$

   Which simplifies to: $$ 4 \times 10^x + 5 \times 10^y = 4.05 $$

4. Guess values for x and y

   Since $10^0 = 1$, $10^{-1} = 0.1$, and $10^{-2} = 0.01$, we will start guessing integer values for $x$ and $y$ that satisfy the equation under $4.05$.

5. Testing some combinations

   Test $x = -1$ and $y = -2$:

   $$ 4 \times 10^{-1} + 5 \times 10^{-2} = 4 \times 0.1 + 5 \times 0.01 $$ $$ = 0.4 + 0.05 = 0.45 $$

   This is too low.

   Now test $x = -1$ and $y = 0$:

   $$ 4 \times 10^{-1} + 5 \times 10^{0} = 0.4 + 5 = 5.4 $$

   This is too high.

   Now let's try $x = -2$ and $y = -1$:

   $$ 4 \times 10^{-2} + 5 \times 10^{-1} = 0.04 + 0.5 = 0.54 $$

   Still too low.

   After a few iterations, we find the combination: $$ x = -1, y = -1 $$

   Replacing these values can lead us back to $0.45$, which allows the equation to hold true.