We can use the point-slope equation:
[tex]y = mx + b[/tex]
m, the slope, is 3/4:
[tex]y = \frac{3}{4} x + b[/tex]
To find b, we plug in the point (4,1/3):
[tex]( \frac{1}{3} ) = \frac{3}{4} (4) + b \\ \frac{1}{3} = 3 + b \\ \frac{1}{3} = \frac{9}{3} + b[/tex]
[tex] - \frac{8}{3} = b[/tex]
Therefore, the point-slope equation is
[tex]y = \frac{3}{4} x - \frac{8}{3} [/tex]
Now we have to see which answer matches.
[tex]y - \frac{3}{4} = \frac{1}{3} (x - 4) \\ y - \frac{3}{4} = \frac{1}{3} x - \frac{4}{3} \\ y - \frac{9}{12} = \frac{1}{3} x - \frac{16}{12}[/tex]
[tex]y = \frac{1}{3} x - \frac{7}{12} [/tex]
Since this is not the same, we try the next one.
[tex]y - \frac{1}{3} = \frac{3}{4} (x - 4) \\ y - \frac{1}{3} = \frac{3}{4} x - 3 \\ y - \frac{1}{3} = \frac{3}{4} x - \frac{9}{3}[/tex]
[tex]y = \frac{3}{4} x - \frac{8}{3} [/tex]
This is the same, so this is the answer. We should still double-check the other answers.
[tex]y - \frac{1}{3} = 4(x - \frac{3}{4} ) \\ y - \frac{1}{3} = 4x - 3 \\ y - \frac{1}{3} = 4x - \frac{9}{3}[/tex]
[tex]y = 4x - \frac{8}{3} [/tex]
This one is not equivalent.
[tex]y - 4 = \frac{3}{4} (x - \frac{1}{3} ) \\ y - 4 = \frac{3}{4} x - \frac{1}{4} \\ y - \frac{16}{4} = \frac{3}{4} x - \frac{1}{4} [/tex]
[tex]y = \frac{3}{4} x + \frac{15}{4} [/tex]
This one also does not work.
The answer is the second one:
[tex]y - \frac{1}{3} = \frac{3}{4} (x - 4)[/tex]
