The slope-intercept form of a line is expressed as follows;
[tex]y=mx+b[/tex]Where;
[tex]\begin{gathered} m=\text{slope} \\ b=y-\text{intercept} \end{gathered}[/tex]To begin, we shall calculate the slope as shown below;
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]Using the two points given, we have;
[tex]\begin{gathered} (x_1,y_1)\Rightarrow(5,7) \\ (x_2,y_2)\Rightarrow(8,22) \end{gathered}[/tex]Therefore;
[tex]\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ m=\frac{22-7}{8-5} \\ m=\frac{15}{3} \\ m=5 \end{gathered}[/tex]To calculate the y-intercept, we shall insert the value of m into the equation, y = mx + b. We shall use the first point which is (5, 7).
Note that if we use the second point (that is 8, 22) the value of the y-intercept would be the same.
Hence we have;
[tex]\begin{gathered} x=5, \\ y=7 \\ m=5 \\ y=mx+b\text{ now becomes;} \\ 7=5(5)+b \\ 7=25+b \end{gathered}[/tex]Subtract 25 from both sides;
[tex]\begin{gathered} -18=b \\ b=-18 \end{gathered}[/tex]Now that we have the values of m and b, (the slope and the y-intercept), the equation becomes;
[tex]\begin{gathered} y=mx+b \\ y=5x+(-18) \\ y=5x-18 \end{gathered}[/tex]ANSWER:
The correct answer is option (b);
[tex]y=5x-18[/tex]