Equation of a Line
Vector Equation
A line is defined as the set of alligned points on the plane with a point, P, and a directional vector, 
.
If P(x1, y1) is a point on the line and the vector 
has the same direction as , then equals multiplied by a scalar unit:
Examples
A line passes through point A = (−1, 3) and has a directional vector with components (2, 5). Determine the equation of the vector.
Write the vector equation of the line which passes through the points A = (1, 2) and B = (−2, 5).
Parametric Form
Examples
A line through point A = (−1, 3) has a direction vector of = (2, 5). Write the equation for this vector in parametric form.
In parametric form, write the equation of the line which passes through the points A = (1, 2) and B = (−2, 5).
Point–Slope Form
m is the slope of the line and (x1, y1) is any point on the line.
Examples
Calculate the point-slope form equation of the line passing through points A = (−2, −3) and B = (4, 2).
Calculate the equation of the line with a slope of 45° which passes through the point (−2, −3).
A line passes through Point A = (−1, 3) and has a direction vector of = (2, 5). Write the equation of the line in point-slope form.
Two-Point Form
As the value of the parameter t from the parametric equations are equal:
The two-point form equation of the line can also be written as:
Example
Determine the equation of the line that passes through the points A = (1, 2) and B = (−2, 5).
General Form
A, B and C are constants and the values of A and B cannot both be equal to zero.
The equation is usually written with a positive value for A.
The slope of the line is:
The direction vector is:
Examples
Determine the equation in general form of the line that passes through Point A = (1, 5) and has a slope of m = −2.
Write the equation in general form of the line that passes through points A = (1, 2) and B = (−2, 5).
Slope–Intercept Form
If the value of y in the general form equation is isolated, the slope–intercept form of the line is obtained:
The coefficient of x is the slope, which is denoted as m.
The independent term is the y-intercept which is denoted as b.
Example
Calculate the equation (in slope–intercept form) of the line that passes through Point A = (1,5) and has a slope m = −2.
Intercept Form
The intercept form of the line is the equation of the line segment based on the intercepts with both axes.
a is the x-intercept.
b is the y-intercept.
a and b must be nonzero.
The values of a and b can be obtained from the general form equation.
If y = 0, x = a.
If x = 0, y = b.
A line does not have an intercept form equation in the following cases:
1.A line parallel to the x-axis, which has the equation y = k.
2.A line parallel to the x-axis, which has the equation x = k.
3.A line that passes through the origin, which has equation y = mx.
Examples
1. A line has an x-intercept of 5 and a y-intercept of 3. Find its equation.
2.The line x − y + 4 = 0 forms a triangle with the axes. Determine the area of the triangle.
The line forms a right triangle with the origin and its legs are the axes.
If y = 0 
x = −4 = a.
If x = 0 y = 2 = b.
The intercept form is:
The area is:
3.A line passes through the point A = (1, 5) and creates a triangle of 18 u² with the axes. Determine the equation of the line.
Apply the intercept form:
The area of the triangle is:
Solve the system:
4.A line forms a triangle with the axes where the length of the leg formed by the x-axis is twice the length of the leg formed by the y-axis. If the line passes through the point A = (3, 2), what is its equation?
