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, Vector.



Direction Vector

If P(x1, y1) is a point on the line and the vector Vector has the same direction as Vector, then Vector equals Vector multiplied by a scalar unit:

Direction Vector

Direction Vector

Direction Vector

Direction Vector

Examples

A line passes through point A = (−1, 3) and has a directional vector Vector with components (2, 5). Determine the equation of the vector.

Vector Equation Example


Write the vector equation of the line which passes through the points A = (1, 2) and B = (−2, 5).

Vector Equation Example

Vector Equation Solution


Parametric Form



Parametric Form

 

Parametric Form Formula

Parametric Form

Parametric Form

Parametric Form

Examples

A line through point A = (−1, 3) has a direction vector of vector = (2, 5). Write the equation for this vector in parametric form.

Parametric Form Solution


In parametric form, write the equation of the line which passes through the points A = (1, 2) and B = (−2, 5).

Parametric Form Calculations

Parametric Form Solution


Point–Slope Form

Point–Slope Form Equation

m is the slope of the line and (x1, y1) is any point on the line.

Slope Equation

Equation for Slope

Slope Formula


Examples

Calculate the point-slope form equation of the line passing through points A = (−2, −3) and B = (4, 2).

Point–Slope Form Example

Point–Slope Form Solution


Calculate the equation of the line with a slope of 45° which passes through the point (−2, −3).

Point–Slope Form Calculations

Point–Slope Form Solution


A line passes through Point A = (−1, 3) and has a direction vector of Vector = (2, 5). Write the equation of the line in point-slope form.

Point–Slope Form Solution


Two-Point Form

As the value of the parameter t from the parametric equations are equal:

Two-point Form

Two-point Form

Two-point Form OperationsTwo-point Form Operations

Two-point Form

The two-point form equation of the line can also be written as:

Two-point Form


Example

Determine the equation of the line that passes through the points A = (1, 2) and B = (−2, 5).

Two-Point Form Solution


General Form

General Form Equation

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:

Slope of a Line

The direction vector is:

Direction Vector


Examples

Determine the equation in general form of the line that passes through Point A = (1, 5) and has a slope of m = −2.

General Form Calculations

General Form Solution


Write the equation in general form of the line that passes through points A = (1, 2) and B = (−2, 5).

General Form Calculations

General Form Solution


Slope–Intercept Form

If the value of y in the general form equation is isolated, the slope–intercept form of the line is obtained:

Slope–Intercept Form

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.

Slope–Intercept Form Calculations

Slope–Intercept Form Solution


Intercept Form

The intercept form of the line is the equation of the line segment based on the intercepts with both axes.

Intercept Form

Intercept Form Equation of a Line

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 flechas x = −4 = a.

If x = 0 flechas y = 2 = b.

Intercept Form

The intercept form is:

Intercept Form Calculations

The area is:

Intercept Form Calculations


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:

Intercept Form Calculations

The area of the triangle is:

Intercept Form Calculations

Solve the system:

Intercept Form Calculations

Intercept Form Calculations

Intercept Form Calculations

Intercept Form Calculations

Intercept Form Solution


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?





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