The distance between the points (1, 3) and (x, 7) is 5, find x.

Distance between 2 points (x₁, y₁) and (x₂, y₂) = $\sqrt{(x$2 - x1)^2 + (y2 - y1)^2}

Distance between (1, 3) and (x, 7) =

$\Rightarrow 5 = \sqrt{(x - 1)^2 + (7 - 3)^2}\\[1em] \Rightarrow 5^2 = (x - 1)^2 + (4)^2\\[1em] \Rightarrow 25 = x^2 + 1 - 2x + 16\\[1em] \Rightarrow x^2 + 1 - 2x + 16 - 25 = 0\\[1em] \Rightarrow x^2 - 2x - 8 = 0\\[1em] \Rightarrow x^2 - 4x + 2x - 8 = 0\\[1em] \Rightarrow x(x - 4) + 2(x - 4) = 0\\[1em] \Rightarrow (x - 4)(x + 2) = 0\\[1em] \Rightarrow x - 4 = 0 \text{ or } x + 2 = 0\\[1em] \Rightarrow x = 4 \text{ or } x = -2\\[1em]$

Hence, the value of x is 4 or -2.