# Find the slope of the linear function if f(2)=-9 and f(3)=9?

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From enunciation, we conclude that we have two points (2,-9) and (3,9) that are located on the graph of the linear function f(x).

We'll write the linear function in the point slope form:

f(x) = mx + n, where m is the slope and n is the y intercept.

If f(2) = -9, we'll substitute x by 2 in the expression of the linear function:

f(2) = 2m + n

2m + n = -9

n = -9 - 2m (1)

If f(3) = 9, we'll substitute x by 3 in the expression of the linear function:

f(3) = 3m + n

3m + n = 9

n = 9 - 3m (2)

We'll put (1) = (2):

-9 - 2m = 9 - 3m

We'll add 3m both sides:

3m - 2m - 9 = 9

We'll add 9 both sides:

m = 9 + 9

**m = 18**

**The slope of the linear function is m = 18.**

A linear function can be written in the form f(x) =ax+b.

Given the values f(2) = -9 and f(3) = 9.

f(-3) = 9 gives a(-3)+b = -9.

-3a+b = -9.....(1).

f(3) = 9 gives: a(3)+b = 9. Or

3a+b = 9.......(2)

Adding (1) and (2) gives:

2b = -9+9 = 0. So b= 0.

(2)-(1) gives: 6a = 9-(-9) = =18.

So 6a = 18.

a = 18/6 = 3.

Therefore f(x) = ax+b = 3x is the required linear function.