To solve the equation \(7 + 4bd = 22 + 9c\) for \(b\), we need to isolate \(b\) on one side of the equation. Here's the step-by-step solution:
1. Begin by moving the constants to one side of the equation and the terms involving \(b\) to the other side:
\(4bd - 9c = 22 - 7\)
Simplifying the right side:
\(4bd - 9c = 15\)
2. Next, factor out \(b\) on the left side:
\(b(4d) - 9c = 15\)
3. Divide both sides of the equation by \(4d\) to solve for \(b\):
\(\frac{{b(4d) - 9c}}{{4d}} = \frac{{15}}{{4d}}\)
Simplifying the left side:
\(b - \frac{{9c}}{{4d}} = \frac{{15}}{{4d}}\)
4. Finally, isolate \(b\) by adding \(\frac{{9c}}{{4d}}\) to both sides of the equation:
\(b = \frac{{15}}{{4d}} + \frac{{9c}}{{4d}}\)
Simplifying the right side:
\(b = \frac{{15 + 9c}}{{4d}}\)
Therefore, the solution for \(b\) is \(b = \frac{{15 + 9c}}{{4d}}\).