Question

find an equation in slope intercept form for the line through (2,-3) and parallel to the line 2x+5y=3

Answer 1

keeping in mind that parallel lines have the same exact slope, then hmmmm what is the slope of 2x+5y = 3 anyway?

$\bf 2x+5y=3\implies 5y=-2x+3\implies y=\cfrac{-2x+3}{5} \\\\\\ y=\stackrel{slope}{-\cfrac{2}{5}}x+\cfrac{3}{5}\impliedby \textit{slope-intercept form}$

so we're really looking for the equation of a line whose slope is -2/5 and runs through 2, -3,

$\bf (\stackrel{x_1}{2}~,~\stackrel{y_1}{-3})\qquad \qquad \qquad slope = m\implies -\cfrac{2}{5} \\\\\\ \stackrel{\textit{point-slope form}}{y- y_1= m(x- x_1)}\implies y-(-3)=-\cfrac{2}{5}(x-2) \\\\\\ y+3=-\cfrac{2}{5}x+\cfrac{4}{5}\implies y=-\cfrac{2}{5}x+\cfrac{4}{5}-3\implies y=-\cfrac{2}{5}x-\cfrac{11}{5}$