# What is the linear function that passes through the points (2,1) and (1,1)?

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A linear function is one where the largest power of x is equal to 1. This condition ensures that the graph of the function is a straight line, hence the term linear function.

Let the linear function be f(x) = y = mx + n

As this passes through (2,1) and (1,1)

y = mx + n

=> 1 = 2m + n

=> n = 1-2m

Also, 1 = m + n

substitute n = 1-2m in this

=> 1 = m + 1 - 2m

=> -2m = 0

=> m = 0

n = 1 - 0 = 1

**Therefore the required linear function is y = 1 or f(x) = 1.**

To find the linear function that passes through the points (2,1) and (1,1).

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

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

So the line that passes through (2,1) and (1,1) is given by:

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

y-1 = (0/-1)(x-2)

y-1 = 0.

So y = 1 is the line that passes through (2,1) and (1,1).

We'll write the form of a linear function:

f(x) = ax + b

A linear function is determined when it's coefficients are determined. So, we'll have to determine the coefficients a and b.

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

f(2) = 1

f(2) = a*2 + b

f(1) = 1

f(1) = a+b

a + b = 1 (2)

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

2a + b = a + b

We'll combine and eliminate like terms:

2a - a = b - b

a = 0

We'll substitute a in (2):

b = 1

**Since the expression of the function is: ****f(x) = 1, the function is not linear, but constant.**