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

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You may use the point slope form of equation of line, such that:

`y - y_0 = f'(x_0)(x - x_0)`

Considering `(x_0,y_0) = (1,2)` and `(x_1,y_1) = (3,5)` yields:

`f'(x_0) = (y_1 - y_0)/(x_1 - x_0)`

`f'(1) = (5 - 2)/(3 - 1) => f'(1) = 3/2`

Hence, you may write the equation of the line passing through the given points, such that:

`y - 2 = (3/2)(x - 1) => y - (3/2)x + 3/2 - 2 = 0`

`-3x + 2y - 1 = 0 => 3x - 2y + 1 = 0`

**Hence, evaluating the equation of the line, under the given conditions, yields **`3x - 2y + 1 = 0.`

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