# Determine the linear function determined by the points (1,2) and (3,5).

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

f(x) = ax + b

Since the function is determined by the points (1,2) and (3,5), 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) = 2

f(1) = a*1 + b

a + b = 2 (1)

f(3) = 5

f(3) = a*3 + b

3a + b = 5 (2)

We'll compute a from the relation (1):

a = 2 - b (3)

We'll substitute (3) in (2):

3(2 - b) + b = 5

We'll remove the brackets and we'll get:

6 - 3b + b - 5 = 0

We'll combine like terms:

-2b + 1 = 0

-2b = -1

We'll divide by -2:

**b = 1/2**

We'll substitute b in (3):

a = 2 - 1/2

a = (4-1)/2

**a = 3/2**

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

**f(x) = 3x/2 + 1/2**

The linear function determined by the points (1,2) and (3,5) is the line passing through these two points .

We know that the line passing through (x1, y1) and (x2, y2) is

y-y1 = {(y2-1)/(x2-x1)} {x-x1).

So the line passing throggh (1,2) and (3,5) is

y- 2 = { (5-2)/(3-1)}(x-1).

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

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

2y-4 = 3x-3.

3x-2y -3+4 = 0.

3x-2y +1 = 0.

Therefore, the linear function determind by the points (1,2) and (3, 5) is 3x-2y +1 = 0.