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Linear Function Formula
Before going to learn the linear function formulas, let us recall what is a linear equation and what is a function. A linear equation is an equation in which every term is either just a constant or the product of a constant and a variable of exponent 1. A function is a relation in which every input has exactly one output. Every linear equation represents a line. All lines except vertical lines are functions. Let us learn linear function formulas along with a few solved examples.
What Are Linear Function Formulas?
There are multiple linear function formulas to find the equation of a line depending on the available information. Linear function formulas are applicable to find the equation of any line except vertical line as vertical line is NOT a function. The linear function formulas are:
 y = mx + b (slopeintercept form)
 \(yy_1=m(xx_1)\) (pointslope form)
 Ax + By = C (standard form)
 \(\dfrac{x}{a}+\dfrac{y}{b}=1\) (intercept form)
Here,
 (x, y) in every equation is a general point on the line.
 \((x_1,y_1)\) is any fixed point on the line.
 m is the slope of the line. It can be defined as \(\dfrac{\text{Rise}}{\text{Run}}\) (or) \(\dfrac{y_2y_1}{x_2x_1}\), where \((x_1, y_1)\) and \((x_2,y_2)\) are any two points on the line.
 (a, 0) and (0, b) are the xintercept and yintercept respectively.
 A, B, and C are constants.
Let us see how to use the linear function formulas in the following solved examples section.
Solved Examples Using Linear Function Formulas

Example 1: Find the linear function of the form y = f(x) that has two points (3, 1) and (2, 4) on it using one of the linear function formulas.
Solution:
The given two points are:
\((x_1,y_1)\) = (3, 1)
\((x_2,y_2)\) = (2, 4)
The slope is, \(m=\dfrac{y_2y_1}{x_2x_1}= \dfrac{4+1}{23}=5\)
The linear function is found using the pointslope form.
\(yy_1=m(xx_1)\)
y + 1 = 5 (x  3)
y + 1 = 5x + 15
y = 5x + 14
Answer: The required linear function is y = 5x + 14.

Example 2 : A car rental in a city rents its cars for an upfront fee of $5 and a daily fee of $30 based on the number of days the car was rented. If x is the number of days and y is the total fee then write the linear relationship between x and y.
Solution:
The upfront fee = $5.
Daily fee = $30.
The total number of days = x.
So the fee for x days = 30x.
Using one of the linear function formulas (slopeintercept formula),
y = mx + b
y = 30x + 5.
Answer: The required linear relationship is y = 30x + 5.
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