\( 1 \leftarrow \quad \) Determine \( b \) so that the slope of \( \overline{\mathrm{PQ}} \) is 6, where \( P \) and \( Q \) are the points \( (b, 4) \) and \( (10,34) \), respectively. What is the value of \( b \) ? \( b=\square \) (Simplify your answer. Type an integer or a fraction.)

To determine \( b \) so that the slope of line segment \( \overline{\mathrm{PQ}} \) is 6, we can use the slope formula: \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \] Here, \( P = (b, 4) \) and \( Q = (10, 34) \). Plugging in the coordinates into the formula, we get: \[ 6 = \frac{34 - 4}{10 - b} \] Simplifying, we have: \[ 6 = \frac{30}{10 - b} \] To eliminate the fraction, we can cross-multiply: \[ 6(10 - b) = 30 \] Now, expanding and solving for \( b \): \[ 60 - 6b = 30 \] Subtracting 60 from both sides gives: \[ -6b = 30 - 60 \] \[ -6b = -30 \] Dividing by -6 gives: \[ b = 5 \] So, the value of \( b \) is \( \boxed{5} \).