How do you write the point slope form of the equation given (-5,-1) and (0,-5)?

2 Answers
Jan 3, 2018

#y+1=-4/5(x+5)color(white)("xxx")orcolor(white)("xxx")y+5=-4/5(x-0)#

Explanation:

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].

Jan 3, 2018

The point-slope form is #y+1=-4/5(x+5)#.

Explanation:

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.