Refer to the attached plot, \[\text{Plot}\left[\left\{15-3x,\frac{9}{5}-\frac{4 x}{5}\right\},\{x,-1,8\}\right] \]
The plot program requires that the two equations be solved for y.
It then computes all of the points displayed. For each value of x between -1 and 8 it then computes the corresponding value for y. This plot program was running under an Apple 64 bit OS and the results are instantaneous.
The fact is that a line is relatively simple to plot for a human because two points determine a line. If you know where the line crosses the x axis and where it crosses the y axis, then all you have to do is mark the two points on graph paper and draw a line through them with a ruler. The math lingo for these points is the x intercept and y intercept respectively. The process has been described fairly well by polpak's second posting.
The expressions in the plot statement are the result of solving for y. If y is zero, in the first expression, then the corresponding x has to be 5 ie: (5,0) If you refer to the plot you will see the blue line crossing the x axis at (5,0).
The blue line appears to cross the y axis at 15 or through the point (0,15). That is easy to confirm because when x is zero, 15 - 3x = 15. For the red line, the y intercept is easy to compute because when x = 0, 9/5 - 4/5 times x = 9/5 = 1.8
The solution to the problem is the point coordinates where they cross each other. In this case by eye ball, x=6 for sure and y is close to -3.