How do you write the equation in point slope form given (3,5) and (8,15)?

If the slope #(m)# of a line and the coordinate #(x_1, y_1)# of one end-point of the line is known, we can write the equation of the line in Point-Slope Form:

#color(red)(y-y_1=m(x-x_1)#

To find the slope if we are given two end-points of a line, we use the formula:

#color(blue)(Slope(m)=(y_2-y_1)/(x_2-x_1)#

We are given the points: #(3,5) and (8,15)#

Note that #x_1 = 3; y_1=5; x_2=8 and y_2=15#

#Slope(m)=(15-5)/(8-3)#

#rArr m=10/5#

#rArr Slope(m)=2#

Next, consider the equation of the point-slope form.

Consider the point #(3,5)#

We also found that #Slope(m)=2#

#x_1=3 and y_1=5#

#y-y_1=m(x-x_1)#

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

#rArr y-5=2x-6#

Add #5# to both sides of the equation.

#rArr y-5+5=2x-6+5#

#rArr y-cancel 5+cancel 5=2x-6+5#

#rArr color(green)(y=2x-1#

This is the required equation in the Point-Slope Form.

We can also graph the line and find the intercepts.

Substitute #y=0# to obtain the x-intercept.

#2x-1=0#

Add #1# to both sides.

#rArr 2x-1+1=0+1#

#rArr 2x-cancel 1+cancel 1=0+1#

#rArr 2x = 1#

Divide both sides by #2#

#(2x)/2=1/2#

#(cancel 2x)/cancel 2=1/2#

#x=1/2#

#x=0.5#

Hence #(0.5,0)# is the x-intercept.

Substitute #x=0# to obtain the y-intercept.

#y=2(0)-1#

#y=0-1#

#y=-1#

Hence #(0,-1)# is the y-intercept.

#(y - y_1) / (y_2 - y_1) = (x - x_1) / (x_2 - x_1)#

#"Given : (x_1, y_1) = (3,5), (x_2,y_2) = (8,15)#

#(y - 5) / (15-5) = (x - 3) / (8 - 3)#

#(y-5) / cancel(10)^color(red)(2) = (x - 3) / cancel 5#

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

#y - y_1 = m * (x - x_1)#

#m = (y_2-y_1)/(x_2-x_1)#

This formula makes sense because slope is rise over run. Rise being the #y# values and run being the #x# values.

Now you choose which point is point 2 (includes #y_2# and #x_2#) and which point is point 1 (includes #y_1# and #x_1#)

Point 2: #(3,5)->y_2 = 5# and #x_2 =3# Point 1: #(8,15)->y_1 = 15# and #x_1 = 8#

#m = (5-15)/(3-8)#

#m = (-10)/(-5)#

#m = (10)/(5)#

#m=2#

Now we know that the slope, #m=2#

#y-y_1=m(x-x_1)#

You can plug in either point for the #x_1# and #y_1# values. Let's use the point #(3,5)#.

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

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

Distribute the #2#

#y-5=2x-6#

Add #5# to both sides

#y-5+5=2x-6+5#

#y=2x-1#

#y=2x-1#