Solution:
The equation of a line that passes through two points is expressed as
[tex]\begin{gathered} y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1) \\ where \\ (x_1,y_1)\text{ and} \\ (x_2,y_2)\text{ are the coordinates of the points } \\ through\text{ which the line passes} \end{gathered}[/tex]
Given that the line passes through the points (2,3) and (2, 5), this implies that
[tex]\begin{gathered} x_1=2 \\ y_1=3 \\ x_2=2 \\ y_2=5 \end{gathered}[/tex]
By substitution, we have
[tex]\begin{gathered} y-3=\frac{5-3}{2-2}(x-2) \\ \Rightarrow y-3=\frac{2}{0}(x-2) \\ multiply\text{ through by zero} \\ 0(y-3)=2(x-2) \\ \Rightarrow0=2x-4 \\ add\text{ 4 to both sides} \\ 0+4=2x-4+4 \\ \Rightarrow4=2x \\ divide\text{ both sides by the coefficient of x, which is 2} \\ \frac{4}{2}=\frac{2x}{2} \\ \Rightarrow x=2 \\ \end{gathered}[/tex]
Hence, the equation of the line that passes through the given points (2,3) and (2,5) is
[tex]x=2[/tex]
