How do you find a standard form equation for the line with (-2,2) and (0,5)?

One way to do this is to find the slope-intercept form of the equation, and then turn that into standard form.

Another way is to use the point-slope form and turn that into standard form.

Using the slope-intercept form of the equation

#y = mx + b#

1) First find slope
slope #= m = (y- y')/(x - x')#

Let #(0,5)# be #(x',y')#
But you can make either point be #(x',y')# and it will come out the same.
I just picked that one because I thought it looked easier to subtract.

Sub in the given values for the variables in the formula
#m# = #(y - y') / (x - x")#

#m =(2 - 5) / (-2-0)#

#m = (-3) / (-2)#

#m = 3/2#

So far, you can write the partial slope intercept formula as

#y = (3/2)x + b#

~ ~ ~ ~ ~ ~ ~ ~ ~

2) Now find b

The formula has 4 unknowns, but you already know 3 of them

#y = 5# (from the given ordered pair)

#m = 3/2# (which you just found)

#x = 0# (from the same ordered pair)

#b# ---- Find #b#

NOTE: You can tell that #5# is the #y# intercept because the point #(0,5)# is the point where #x# is zero.
This is the place where the line crosses the y axis.

But if you didn't see that, here is how to calculate #b#

Sub in the values into the formula and solve for #b#

#y = mx + b#

#5 = (3/4)*0 + b#

#5 = 0 + b#

#5 = b# #larr# #b# is the #y# intercept

3) Now you can write the whole equation

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

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The question wants this equation expressed in standard form.

Standard form is
#ax + by = c#
where a is a positive whole integer

Start with the slope intercept form.

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

4) Now put the formula in standard form.

1) Clear the fraction by multiplying every term on both sides by #2# and letting the denominator cancel
#2y = 3x + 10#

2) Subtract #10# from both sides
#2y - 10 = 3x#

3) Subtract #2y# from both sides
#3x - 2y = - 10# #larr# standard form