# What is the linear function f if the graph of f passes through the point (1;3) and (2;1) ?

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A linear function is determined when it's coefficients are determined.

y = f(x) = mx + n

So, in order to determine y, we'll have to calculate the coefficients m and n.

Since the function is determined by the points (1,3) and (2,1), that means that if we'll substitute the coordinates of the points into the expression of the function, we'll get the relations:

f(1) = 3

f(1) = m*1 + n

m + n = 3 (1)

f(2) = 1

f(2) = 2m + n

2m + n = 1 (2)

We'll subtract (1) from (2):

2m + n -m - n = 1 - 3

We'll combine and eliminate like terms:

**m = -2**

We'll substitute m in (1):

m + n = 3

-2 + n = 3

n = 3+2

**n = 5**

**The expression of the linear function is:**

**f(x) = -2x + 5**

The linear function or the line that passes through the two points (x1,y1) and (x2,y2) is given by:

y-y1 = {(y2-y1)/(x2-x1)}(x-x1).

So the linear function that passes through (the points (1,3) and (2,1) is given by:

y-3 = {((1-3)/((2-1)}(x-1).

y-3 = (-2/-1)(x-1)

y-3 = 2(x-1).

y-1 = 2x-2.

2x-y-2+1 = 0

2x-y-1 = 0 .

Therefore the linear function that passes through the points (1,3) and (2,1) is 2x-y-1 = 0.