We can determine the fundamental equation of a line using the angle formed by the line with the abscissa axis (x) and the coordinates of a point belonging to the line. The angular coefficient of the line, associated with the coordinate of the point, facilitates the representation of the equation of the line. Watch:
Considering a line r, the point C(xÇyÇ) belonging to the line, its slope m and another generic point D(x, y) different from C. With two points belonging to the line r, one real and the other generic, we can calculate its slope. 
m = y - y0/x - x0
m (x - x0) = y - y0
Therefore, the fundamental equation of the line will be determined by the following expression:
y-y0 = m (x - x0)
Example 1
Find the fundamental equation of the line r that has the point A (0,-3/2) and slope equal to m = – 2.
y – y0 = m (x – x0)
y – (–3/2) = –2(x – 0)
y + 3/2 = –2x
2x + y + 3/2 = 0
Example 2
Obtain an equation for the line shown below: 
To determine the fundamental equation of the line we need the coordinates of one of the points belonging to the line and the value of the slope. The coordinates of the given point is (5,2), the slope is the tangent of the angle α.
We will obtain the value of α with the difference 180° – 135° = 45°, so α = 45° and a tg 45° = 1.
y-y0 = m (x - x0)
y – 2 = 1 (x – 5)
y – 2 = x – 5
y - x + 3 = 0
Example 3
Find the equation of the line passing through the coordinate point (6; 2) and has a 60º inclination.
Angular coefficient is given by the tangent of the 60º angle: tg 60º = √3.
y-y0 = m (x - x0)
y – 2 = √3 (x – 6)
y – 2 = √3x – 6√3
–√3x + y – 2 + 6√3 = 0
√3x – y + 2 – 6 √3 = 0
by Mark Noah
Graduated in Mathematics
Brazil School Team
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Analytical Geometry - Math - Brazil School
Would you like to reference this text in a school or academic work? Look:
SILVA, Marcos Noé Pedro da. "Fundamental Equation of the Line"; Brazil School. Available in: https://brasilescola.uol.com.br/matematica/equacao-fundamental-reta-1.htm. Accessed on June 28, 2021.
