To solve the exercise, we can first find the slope of the line using this formula:
[tex]\begin{gathered} m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ \text{ Where m is the slope and} \\ (x_1,y_1)\text{ and }(x_2,y_2)\text{ are two points through which the line passes} \end{gathered}[/tex]
So, in this case, we have:
[tex]\begin{gathered} (x_1,y_1)=(3,3) \\ (x_2,y_2)=(4,5) \\ m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ m=\frac{5-3}{4-3} \\ m=\frac{2}{1} \\ $\boldsymbol{m=2}$ \end{gathered}[/tex]
Now, we can use the point-slope formula to find the equation of the line in its slope-intercept form:
[tex]\begin{gathered} y-y_1=m(x-x_1)\Rightarrow\text{ Point-slope formula} \\ y-3=2(x-3) \\ \text{ Apply the distributive property from both sides of the equation} \\ y-3=2\cdot x-2\cdot3 \\ y-3=2x-6 \\ \text{ Add 3 from both sides of the equation} \\ y-3+3=2x-6+3 \\ y=2x-3 \end{gathered}[/tex]
Therefore, the equation of the line passing through the points (3,3) and (4,5) is
[tex]$\boldsymbol{y=2x-3}$[/tex]
