https://file.aiquickdraw.com/mathbot/file/17267760420803vy2irnf.jpg

The equation provided in the image is:

\[
\frac{7}{4d + 1} = \frac{3}{2d - 9}
\]

### Steps to solve:

1. **Cross-multiply**:
   Multiply both sides of the equation by \((4d + 1)(2d - 9)\) to eliminate the denominators:

   \[
   7(2d - 9) = 3(4d + 1)
   \]

2. **Distribute**:
   Now, distribute both sides:

   \[
   7 \times 2d - 7 \times 9 = 3 \times 4d + 3 \times 1
   \]

   Simplify:

   \[
   14d - 63 = 12d + 3
   \]

3. **Move terms involving \(d\) to one side**:
   Subtract \(12d\) from both sides:

   \[
   14d - 12d - 63 = 3
   \]

   Simplify:

   \[
   2d - 63 = 3
   \]

4. **Solve for \(d\)**:
   Add 63 to both sides:

   \[
   2d = 66
   \]

   Divide by 2:

   \[
   d = 33
   \]

### Final answer:
\[
d = 33
\]

Would you like more details or have any further questions? Here are some related questions to explore:

1. What happens if we substitute \(d = 33\) back into the original equation?
2. How does cross-multiplication help in solving rational equations?
3. Could there be any extraneous solutions in this problem?
4. How do we handle cases where the equation has no solutions?
5. What if the equation involved more complicated fractions?

**Tip:** Always check for any restrictions on the variable that could make the denominator zero when solving rational equations!

Solve the equation \frac{7}{4d + 1} = \frac{3}{2d - 9}

Learn how to solve the rational equation \frac{7}{4d + 1} = \frac{3}{2d - 9} step-by-step. This algebra problem uses cross-multiplication and linear solving techniques. Suitable for students in Grades 8-10.

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