To solve the equation \( b - 5 + 7b = 24 \), we can first combine like terms.
Combine \( b \) and \( 7b \):
\[ b + 7b = 8b \]
Now, rewrite the equation:
\[ 8b - 5 = 24 \]
Next, add 5 to both sides:
\[ 8b - 5 + 5 = 24 + 5 \]
This simplifies to:
\[ 8b = 29 \]
Now, divide both sides by 8:
\[ b = \frac{29}{8} \]
Now we can check our solution by substituting \( b = \frac{29}{8} \) back into the original equation:
Starting with the left side of the equation:
\[ b - 5 + 7b = \frac{29}{8} - 5 + 7 \cdot \frac{29}{8} \]
First, compute \( 5 \) in terms of eighths:
\[ 5 = \frac{40}{8} \]
Now substitute:
\[ \frac{29}{8} - \frac{40}{8} + 7 \cdot \frac{29}{8} \]
Calculate \( \frac{29}{8} - \frac{40}{8} \):
\[ \frac{29 - 40}{8} = \frac{-11}{8} \]
Now calculate \( 7 \cdot \frac{29}{8} \):
\[ \frac{203}{8} \]
Now combine \( -\frac{11}{8} \) and \( \frac{203}{8} \):
\[ -\frac{11}{8} + \frac{203}{8} = \frac{-11 + 203}{8} = \frac{192}{8} = 24 \]
Since the left-hand side equals 24, our solution is verified:
\[ b = \frac{29}{8} \] is the correct answer.