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.

Linear Equation - dCode

Tag(s) : Geometry, Functions

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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.

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 $

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 $

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`

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

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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Cite as source (bibliography):

*Linear Equation* on dCode.fr [online website], retrieved on 2023-10-01,

- Linear Equation Calculator (2D plane)
- What is a linear equation? (Definition)
- How to calculate the equations of a line?
- How to calculate slope coefficient?
- How to calculate the y intercept?
- How to find the equation of a line from two points?
- How to find the equation of a line from one point and the slope?

line,linear,equation,coefficient,slope,abcissa,ordinate,origin,affine,function

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