On the coordinate plane, point \( P \) is located at \( (3, y) \) and point \( Q \) is located at \( (1,-4) \). Th distance between points \( P \) and \( Q \) is \( \sqrt{29} \) units. What are the two possible values of \( y \) ? \[ y=\square \text { or } y=\square \]

To find the values of \( y \), we can use the distance formula between points \( P(3, y) \) and \( Q(1, -4) \). The formula for distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Plugging in the points \( P \) and \( Q \): \[ \sqrt{(1 - 3)^2 + (-4 - y)^2} = \sqrt{29} \] Calculating the left side: \[ \sqrt{(-2)^2 + (-4 - y)^2} = \sqrt{29} \] \[ \sqrt{4 + (-4 - y)^2} = \sqrt{29} \] Squaring both sides: \[ 4 + (-4 - y)^2 = 29 \] \[ (-4 - y)^2 = 25 \] Taking the square root of both sides gives: \[ -4 - y = 5 \quad \text{or} \quad -4 - y = -5 \] Solving the first equation: \[ -4 - y = 5 \implies -y = 9 \implies y = -9 \] Solving the second equation: \[ -4 - y = -5 \implies -y = -1 \implies y = 1 \] Thus, the two possible values of \( y \) are: \[ y = -9 \text{ or } y = 1 \]