Answer
Point-Slope Form: $y-3=2(x-2)$
Standard Form: $2x-y=1$
Work Step by Step
With the given points, $ (2,3)$ and $(3,5),$ then \begin{align*} y_1&= 3 ,\\y_2&= 5 ,\\x_1&= 2 ,\text{ and }\\ x_2&= 3 .\end{align*} Using $m=\dfrac{y_1-y_2}{x_1-x_2}$ or the Slope Formula, the slope, $m,$ of the line connecting the two given points is \begin{align*} m&=\dfrac{3-5}{2-3} \\\\&= \dfrac{-2}{-1} \\\\&= 2 .\end{align*} Using $ y-y_1=m(x-x_1) $ or the Point-Slope Form, with $m=2$ and using the point $(2,3)$, then \begin{align*} y-3=2(x-2) .\end{align*} A Point-Slope Form of the line with the given conditions is $ y-3=2(x-2) .$ In the form $ax+by=c$ or the Standard Form, the equation above is equivalent to \begin{align*} y-3&=2(x)-2(2) \\ y-3&=2x-4 \\ -2x+y&=-4+3 \\ -2x+y&=-1 \\ -1(-2x+y)&=(-1)(-1) \\ 2x-y&=1 .\end{align*} The standard form of the line with the given conditions is $ 2x-y=1 .$
