dimance between \( (1, y) \) and \( (5,4) \) is 5 , where \( y>4 \), then \( y \) is equat to

To find the value of \( y \), we start by using the distance formula between the points \( (1, y) \) and \( (5, 4) \), which is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Here, \( d = 5 \), \( (x_1, y_1) = (1, y) \), and \( (x_2, y_2) = (5, 4) \). Plugging in these values, we have: \[ 5 = \sqrt{(5 - 1)^2 + (4 - y)^2} \] This simplifies to: \[ 5 = \sqrt{16 + (4 - y)^2} \] Squaring both sides: \[ 25 = 16 + (4 - y)^2 \] Subtracting 16 from both sides gives: \[ 9 = (4 - y)^2 \] Taking the square root of both sides results in: \[ 4 - y = 3 \quad \text{or} \quad 4 - y = -3 \] From \( 4 - y = 3 \): \[ y = 1 \] From \( 4 - y = -3 \): \[ y = 7 \] Since we require \( y > 4 \), we select \( y = 7 \). Therefore, the value of \( y \) is \( \boxed{7} \).