How do you write the equation in slope intercept form given (1,4) and (2,-5)?

Let's find the slope for #y=color(red)(m)x+b#, using #(y_2 - y_1)/(x_2 - x_1)#

#(-5 - 4)/(2 - 1)# gives us #-9/1# for #m#

Now we need to find #b#, which is the #y#-intercept. That means, the value of #y# when #x=0#

To find that, let's us the point-slope formula again:

#(y_2 - y_1)/(x_2 - x_1)# with #(0,?)# and #(1, 4)#

#(? - 4)/(0 - 1)#
Now what? Well, we already know what the slope is. It's #-9/1#. Now we can solve!

#(? - 4)/(0 - 1)=-9/1#

#(?-4)/-1=-9#

multiply by #-1# on both sides

#?-4=9#

add #4# on both sides

#?=13#

Now we have our #y#-intercept. It's #13#!

#y=-9x+13#

Let's graph our equation and make sure it goes through the points #(2, -5)# and #(1, 4)#:

graph{y=-9x+13}
It does! We were right

#"the equation of a line in "color(blue)"slope-intercept form"# is.

#color(red)(bar(ul(|color(white)(2/2)color(black)(y=mx+b)color(white)(2/2)|)))#
where m represents the slope and b. the y-intercept.

#"to calculate m use the "color(blue)"gradient formula"#

#color(red)(bar(ul(|color(white)(2/2)color(black)(m=(y_2-y_1)/(x_2-x_1))color(white)(2/2)|)))#
# (x_1,y_1),(x_2,y_2)" are 2 coordinate points"#

#"the points are " (x_1,y_1)=(1,4),(x_2,y_2)=(2,-5)#

#rArrm=(-5-4)/(2-1)=(-9)/1=-9#

#rArry=-9x+blarr" is the partial equation"#

#"to find b, substitute either of the 2 given points into"#
#"the partial equation"#

#"using " (1,4)#

#4=-9+brArrb=13#

#rArry=-9x+13larrcolor(red)" in slope-intercept form"#