Answer:
[tex]y=\frac{2}{3} x+\frac{7}{3}[/tex]
Step-by-step explanation:
Parallel lines have the same slope. Therefore, the line we're trying to find the equation for and the given line 2x-3y=1 must have the same slope.
1) Find the slope of 2x-3y=1
To do this, rewrite this equation in slope-intercept form: [tex]y=mx+b[/tex] where [tex]m[/tex] is the slope and [tex]b[/tex] is the y-intercept (the value of y when the line crosses the y-axis)
[tex]2x-3y=1[/tex]
Subtract 2x from both sides
[tex]2x-3y-2x=1-2x\\-3y=-2x+1[/tex]
Divide both sides by -3 to isolate y
[tex]\frac{-3y}{-3}=\frac{-2}{-3}x+(\frac{1}{-3} )\\y=\frac{2}{3}x-\frac{1}{3}[/tex]
Now, we can easily identify that [tex]\frac{2}{3}[/tex] is in the position of m, the slope. Plug this into [tex]y=mx+b[/tex]:
[tex]y=\frac{2}{3} x+b[/tex]
2) Find the y-intercept
[tex]y=\frac{2}{3} x+b[/tex]
To find the y-intercept, plug the given point P(-2,1) into the equation and solve for b
[tex]1=\frac{2}{3} (-2)+b\\1=\frac{-4}{3}+b[/tex]
Add [tex]\frac{4}{3}[/tex] to both sides of the equation
[tex]1+\frac{4}{3}= \frac{-4}{3} +b+\frac{4}{3} \\\frac{3}{3} +\frac{4}{3}= b\\\frac{7}{3}[/tex]
Therefore, the y-intercept of the line is [tex]\frac{7}{3}[/tex]. Plug this into our original equation:
[tex]y=\frac{2}{3} x+b\\y=\frac{2}{3} x+\frac{7}{3}[/tex]
I hope this helps!
