How do you find the slope given #5y - 2x = -3#?

Given the equation of a line, all we need to do is rearrange it into terms of #y=mx+b#

#5y-2x=-3#
#5y=2x-3# Add -2x to both sides to get #y# by itself
#y=2/5x-3/5# Divide all terms by 5

Now that the equation is in terms of slope-intercept, with the slope being #m# in #y=mx+b#, you can find the slope.

We can multiply each side of the equation by #color(red)(-1)# to put the equation in Standard Linear Form. The standard form of a linear equation is: #color(red)(A)x + color(blue)(B)y = color(green)(C)#

Where, if at all possible, #color(red)(A)#, #color(blue)(B)#, and #color(green)(C)#are integers, and A is non-negative, and, A, B, and C have no common factors other than 1

#color(red)(-1)(5y - 2x) = color(red)(-1) * -3#

#(color(red)(-1) xx 5y) - (color(red)(-1) xx 2x) = 3#

#-5y - (-2x) = 3#

#-5y + 2x = 3#

#color(red)(2)x + color(blue)(-5)y = color(green)(3)#

The slope of an equation in standard form is: #m = -color(red)(A)/color(blue)(B)#

Substituting gives:

#m = (-color(red)(2))/color(blue)(-5) = 2/5#

So you're going to want to get it into #mx+b=y# form, where #m# is the slope and #b# is the #x# intercept.

To rearrange the equation:
#5y-2x=-3#
add #2x# to each side, which cancels out #-2x# from the left side
#5y=-3+2x#
now divide each side by #5#, which crosses out the #5# in #5y#
#y=(-3+2x)/5#

You now have the correct arrangement of the equation and can even flip #-3# and #2x# to match the form of the equation you want it in

#y=(2x-3)/5#

Now since you have the equation being divided by #5#, you have to divide both #2# and #3# by #5#, making your new equation:
#y=(2/5)x-(3/5)#

and following the equation we can now see that #m#, which is the slope, is equal to #2/5#.