What is the Value of \( P \), if the gradent of the line joining \( [1, p] \) and \( [p, 4] \) is \( 2 / 3 \)

To find the value of \( p \) given that the gradient (slope) of the line joining the points \( (1, p) \) and \( (p, 4) \) is \( \frac{2}{3} \), follow these steps: 1. **Understand the Slope Formula:** The slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] 2. **Apply the Given Points:** Substitute the given points \( (1, p) \) and \( (p, 4) \) into the slope formula: \[ \frac{4 - p}{p - 1} = \frac{2}{3} \] 3. **Solve for \( p \):** Multiply both sides of the equation by \( 3(p - 1) \) to eliminate the denominator: \[ 3(4 - p) = 2(p - 1) \] Simplify the equation: \[ 12 - 3p = 2p - 2 \] Combine like terms: \[ 12 + 2 = 2p + 3p \] \[ 14 = 5p \] \[ p = \frac{14}{5} \] 4. **Final Answer:** The value of \( p \) is \( \frac{14}{5} \) or \( 2.8 \). **Answer:** The value of \( p \) is fourteen fifths, that is \( p = \dfrac{14}{5} \).