The equation of a line with slope m and y-intercept b in slope-intercept form is:
[tex]y=mx+b[/tex]The slope represents the rate of change of the variable y with respect to the variable x, and the y-intercept represents the initial value of y when x=0.
In this case, let c represent the number of clients as a function of time, and let t represent time in years.
Then, c=46 when t=1 and c=58 when t=2.
Use the slope formula to find the slope of the line that passes through the points (1,46) and (2,58) in a c vs t graph:
[tex]\begin{gathered} m=\frac{c_2-c_1}{t_2-t_1} \\ =\frac{58-46}{2-1} \\ =\frac{12}{1} \\ =12 \end{gathered}[/tex]Replace 12 for the slope and substitute a pair of corresponding values of c and t into the equation to find the initial value. For instance, substitute t=1 and c=46:
[tex]\begin{gathered} c=12t+b \\ \Rightarrow46=12(1)+b \\ \Rightarrow46-12=b \\ \Rightarrow34=b \\ \therefore b=34 \end{gathered}[/tex]To find the equation that gives the number of clients the lawyer will have as a function of time, replace 12 for the slope and 34 for the initial value:
[tex]c=12t+34[/tex]We can verify that we obtain the correct values for c when t=1 and t=2:
[tex]\begin{gathered} c_1=12(1)+34=12+34=46 \\ c_2=12(2)+34=24+34=58 \end{gathered}[/tex]Therefore, the equation that gives the number of clients (c) the lawyer will have (t) years after beginning the firm, is:
[tex]c=12t+34[/tex]
