Point Slope Equation

Question

(-4,8)(4,4)

y-4=

slope = -1/2

y-4= -1/2 (x-4)

Question: I understand how to arrive at slope, but how do I determine (x-4) from this information?

Answer 1

Given: (4,4) and (-4,8) are two points on the line

Then the slope is m=(8-4)/(-4 - 4) = 4/(-8) = -1/2 as given

Point slope form is DEFINED as:

y - y0 = m ( x - x0) where m is the slope and (x0,y0) is a point on the line.

So then using (4,4) as the point on the line, the equation in point slope form is

y-4 = (-1/2)(x-4) where (x0,y0) = (4,4) and m= -1/2

Using the other point (x0,y0) = (-4,8) the equation in point slope form is

y - 8 = (-1/2)(x-(-4))

y - 8 = (-1/2)(x+4)

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To convince yourself these two equations are really the same:

starts with y - 4 = (-1/2)(x-4)

Subtracting 4 from both sides:

y-4-4 = (-1/2)(x-4)-4

y - 8 = (-1/2)(x-4) - 4 <--- combines like terms

= (-1/2)x + 2 - 4 distributes -1/2: signs inside parenthesis change and all numbers cut in half

= (-1/2)x - 2 : <--- combines like terms again

= (-1) * [(1/2)x + 2] <--- factors out -1

= (-1) * (1/2) [ x + 4] <--- factors out 1/2 : numbers inside parenthesis double and switch signs

= (-1/2) (x+4) which is the same as the second equation

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Optionally, you can rewrite both equations in slope intercept form....

y - 4 = (-1/2)(x-4) = (-1/2)x + 2 <--- distributive

Adding 4 to both sides y = (-1/2)x + 6

Now again with y - 8 = (-1/2)(x+4)

= (-1/2)x - 2 <--- distributive

adding 8 to both sides:

y = (-1/2)x + 6

and the slope intercept form is the same for both.