We know that one of the point of our line is (5,-1). Since our second point has the same y-intercept as [tex]x-3y=6[/tex], we are going to solve for x in that line to identify the y-intercept:
[tex]x-3y=6[/tex]
[tex]-3y=6-x[/tex]
[tex]y= \frac{6-x}{-3} [/tex]
[tex]y= \frac{1}{3}x-2 [/tex]
Remember that in a line of the form [tex]y=mx+b[/tex] the intercept is (0,b). From this we can infer that the second point of our line is (0,-2)
Now that we have our two point, we can use the slope formula: [tex]m= \frac{y_{2}-y_{1}}{x_{2}-x_{1} } [/tex] to find the slope of our line:
[tex]m= \frac{-2-(-1)}{0-5} [/tex]
[tex]m= \frac{-2+1}{-5} [/tex]
[tex]m= \frac{-1}{-5} [/tex]
[tex]m= \frac{1}{5} [/tex]
We can conclude that the slope of our line is [tex]m= \frac{1}{5} [/tex], so the correct answer is C.
