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Linear Equation

Tool to calculate a linear equation from 1 or 2 points. The equation of a line is written ax + b with a, the direction coefficient (or slope) and b, the y-intercept.

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Linear Equation -

Tag(s) : Geometry, Functions

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# Linear Equation

## Answers to Questions (FAQ)

### What is a linear equation? (Definition)

A linear equation is a mathematical relationship that describes a straight line in a 2D plane. It is usually in the form $y = ax + b$ (or $f(x) = ax + b$), where $a$ and $b$ are real numbers and $x$ is the variable.

### How to calculate the equations of a line?

The equation on the right, of the form $a x + b$ (affine function), has 2 parameters: $a$ its slope coefficient and $b$ the ordinate at the origin.

Knowing the slope coefficient and the ordinate at the origin, is deduced the linear equation of the line.

Example: A line of slope coefficient 2 and intercept 3 has the equation $2x + 3$

### How to calculate slope coefficient?

From 2 points $A(x_A, y_A)$ and $B(x_B, y_B)$, the formula for the slope coefficient of the line or segment passing through the points A and B is the result of the fraction: $$\frac{y_B-y_A}{x_B-x_A}$$

Example: A line passes through 2 points A(1,2) and B(3,4), the slope coefficient is $\frac{4-2}{3-1} = \frac{2}{2} = 1$

### How to calculate the y intercept?

From the slope coefficient $a$ and a point $A(x_A, y_A)$, the computation formula of the y-intercept is the result $b$ of the equation $$a x_A + b = y_A$$

Example: A line with slope coefficient $3$ goes through A(2,4), so $3 \times 2 + b = 4 \iff b = -2$, so the y intercept is -2

### How to find the equation of a line from two points?

It is possible to find the equation directly from the formula: $$y = \frac{ (x - x_A)(y_B - y_A) }{ (x_B - x_A) } + y_A$$

### How to find the equation of a line from one point and the slope?

For a point $P = (x_1, y_1)$ and a slope $S$, the calculation formula is: $$y = S(x - x_1) + y_1$$

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