How do you write the point slope form of the equation given (-5,-1) and (0,-5)?
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#y+1=-4/5(x+5)color(white)("xxx")orcolor(white)("xxx")y+5=-4/5(x-0)#
Note that the slope between the points #(-5,-1)# and #(0,-5)# is
#color(white)("XXX")color(green)m=(Deltay)/(Deltax)=(-1-(-5))/(-5-0)=color(green)(-4/5)#
The general slope-point form for a line with slope #color(green)m# through the point #(color(blue)(hatx),color(red)(haty))# is
#color(white)("XXX")y-color(red)(haty)=color(green)m(x-color(blue)(hatx))#
We can use either of the given points for our #(color(blue)(hatx),color(red)(haty))#.
If, for example, we use #(color(blue)(hatx),color(red)(haty))=(color(blue)(-5),color(red)(-1))#
then our slope-point form becomes
#color(white)("XXX")ycolor(red)(+1)=color(green)(-4/5)(xcolor(blue)(+5))#
[using #(color(blue)(hatx),color(red)(haty))=(color(blue)(0),color(red)(-5))# gives the alternate, but equivalent, version shown in the Answer].
The point-slope form is #y+1=-4/5(x+5)#.
Slope
You first need to determine the slope from the given points. The formula for determining slope is:
#m=(y_2-y_1)/(x_2-x_1)#,
where:
#m# is the slope, #(x_1,y_1)# and #(x_2,y_2)# are the two points.
Plug the known values into the formula. I'm going to use #(-5,-1)# for Point 1 and #(0,-5)# for Point 2. It doesn't matter which point you make 1 or 2. The result will be the same.
#m=(-5-(-1))/(0-(-5))#
Simplify.
#m=(-5+1)/(0+5)#
Solve.
#m=-4/5#
Point-slope form
#y-y_1=m(x-x_1)#,
where:
#m=-4/5#, #(x_1,y_1)# is one of the points. It doesn't matter which one. I'm going to use #(-5,-1)#.
Plug in the known values.
#y-(-1)=-4/5(x-(-5))#
#y+1=-4/5(x+5)# #larr# Point-slope form
You can solve for #y# to convert it into the slope-intercept form:
#y=mx+b#,
where:
#m# is the slope, #-4/5#, and #b# is the y-intercept.
#y+1=-4/5(x+5)#
Expand the right-hand side.
#y+1=-4/5x-4#
Subtract #-1# from both sides.
#y+1=-4/5x-4-1#
Simplify.
#y=-4/5x-5# #larr# Slope-intercept form.
