Equation of a Line Calculator

How to Calculate the Equation of a Line

Calculating the equation of a line is a fundamental skill in algebra and coordinate geometry. Here's a comprehensive guide on how to find the equation of a line using different methods:

Line Equation Formulas

There are three common forms of a line equation:

1. Slope-Intercept Form: \( y = mx + b \)
   • m is the slope of the line
   • b is the y-intercept (where the line crosses the y-axis)
2. Point-Slope Form: \( y - y_1 = m(x - x_1) \)
   • (x₁, y₁) is a point on the line
   • m is the slope of the line
3. Standard Form: \( Ax + By = C \)
   • A, B, and C are constants
   • A and B are not both zero

Calculation Steps

The steps to calculate the equation of a line depend on the given information:

Using Two Points

1. Calculate the slope using the slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
2. Use the point-slope form with one of the points: \( y - y_1 = m(x - x_1) \)
3. Expand and simplify to get the slope-intercept form: \( y = mx + b \)

Using Slope and a Point

1. Use the point-slope form: \( y - y_1 = m(x - x_1) \)
2. Expand and simplify to get the slope-intercept form: \( y = mx + b \)

Using Slope and Y-Intercept

1. The equation is already in slope-intercept form: \( y = mx + b \)
2. Substitute the given values for m (slope) and b (y-intercept)

Example Calculation

Let's find the equation of a line passing through the points (1, 2) and (3, 6).

1. Calculate the slope: \[ m = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2 \]
2. Use the point-slope form with (1, 2): \[ y - 2 = 2(x - 1) \]
3. Expand the equation: \[ y = 2x - 2 + 2 \]
4. Simplify to get the slope-intercept form: \[ y = 2x \]

Visual Representation

A visual representation can help understand the concept better. Here's a diagram showing the line y = 2x passing through the points (1, 2) and (3, 6):

This diagram illustrates how the line y = 2x passes through the points (1, 2) and (3, 6), confirming our calculated equation.