Find 𝑓(𝑥) if 𝑓 is a linear function such that 𝑓(5)=5 and 𝑓(−4)=−7. 𝑓(𝑥)=

Question

I need to understand the steps on how to find this equation

Answer 1

(5,5), (-4,-7) have a line through them with slope =

(5+7)/(5+4) = 12/9 = 4/3

the equation is

y-5 = (4/3)(x-5) in point slope form

or

y = (4/3)x -5/3 in slope intercept form

4x-3y = 5 in standard form

Answer 2

If f(5) = 5 then the point (5, 5) is on the line. If f(-4) = -7, then the point (-4, -7) is on the line.

The slope equation is:

so the slope (m) = (-7 - 5)/(-4 - 5) = -12/-9 = 4/3

The point-slope form of a line is:

plugging in m = 4/3, x1 = 5 and y1 = 5 we get:

y - 5 = (4/3)(x - 5)

Now, turn it into a function be manipulating it to say y = (something)

y - 5 = (4/3)x - 20/3

y = (4/3)x - 20/3 + 5

y = (4/3)x - 20/3 + 15/3

y = (4/3)x - 5/3

Then replace "y" with "f(x):

f(x) = (4/3)x - 5/3

Answer 3

Start with two-point form (Algebra 1) using the two points (5, 5) and (-4, -7)

(y - 5) / (x - 5) = (-7 -5) / (-4 - 5)