Equation of a Line

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} m&=-\frac ab,\\ x_0&=-\frac ca,\\ y_0&=-\frac cb. \end{align}}

Point–slope form or Point-gradient form

A non-vertical line can be defined by its slope $m$ , and the coordinates Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1, y_1} of any point of the line. In this case, a linear equation of the line is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=y_1 + m(x-x_1),}

or

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=mx +y_1-mx_1.}

This equation can also be written

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y-y_1=m(x-x_1)}

for emphasizing that the slope of a line can be computed from the coordinates of any two points.

Intercept form

A line that is not parallel to an axis and does not pass through the origin cuts the axes in two different points. The intercept values $x Template:Sub$ and $y Template:Sub$ of these two points are nonzero, and an equation of the line is^[2]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{x}{x_0} + \frac{y}{y_0} = 1.}

(It is easy to verify that the line defined by this equation has $x Template:Sub$ and $y Template:Sub$ as intercept values).

Two-point form

Given two different points $(x Template:Sub, y Template:Sub)$ and $(x Template:Sub, y Template:Sub)$ , there is exactly one line that passes through them. There are several ways to write a linear equation of this line.

If $x Template:Sub \neq x Template:Sub$ , the slope of the line is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{y_2 - y_1}{x_2 - x_1}.} Thus, a point-slope form is^[2]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1).}

By clearing denominators, one gets the equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_2 - x_1)(y - y_1) - (y_2 - y_1)(x - x_1)=0,}

which is valid also when $x Template:Sub = x Template:Sub$ (for verifying this, it suffices to verify that the two given points satisfy the equation).

This form is not symmetric in the two given points, but a symmetric form can be obtained by regrouping the constant terms:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (y_1-y_2)x + (x_2-x_1)y + (x_1y_2 - x_2y_1) =0}

(exchanging the two points changes the sign of the left-hand side of the equation).

Determinant form

The two-point form of the equation of a line can be expressed simply in terms of a determinant. There are two common ways for that.

The equation $(x_{2}-x_{1})(y-y_{1})-(y_{2}-y_{1})(x-x_{1})=0$ is the result of expanding the determinant in the equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{vmatrix}x-x_1&y-y_1\\x_2-x_1&y_2-y_1\end{vmatrix}=0.}

The equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (y_1-y_2)x + (x_2-x_1)y + (x_1y_2 - x_2y_1)=0} can be obtained be expanding with respect to its first row the determinant in the equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{vmatrix} x&y&1\\ x_1&y_1&1\\ x_2&y_2&1 \end{vmatrix}=0.}

Beside being very simple and mnemonic, this form has the advantage of being a special case of the more general equation of a hyperplane passing through $n$ points in a space of dimension $n - 1$ . These equations rely on the condition of linear dependence of points in a projective space.

Licensing

Content obtained and/or adapted from:

Linear equation, Wikipedia under a CC BY-SA license

Resources

Equation of a Line, Khan Academy
Slope-intercept equation from slope & point, Khan Academy

↑ Template:Harvnb

↑ ^2.0 ^2.1 Template:Harvnb

[1]

[2]

Equation of a Line

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