# 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(x_{1}, y_{1}) 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 (x_{1}, y_{1}) 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?