Equation of a straight line

Equation of a Straight Line

The equation of a straight line is a fundamental concept in analytical geometry. Understanding it allows you to describe the relationship between two variables in a two-dimensional space. The general form of the equation of a straight line is:

y = mx + c

In this equation:

• y is the dependent variable.
• x is the independent variable.
• m is the gradient (slope) of the line.
• c is the y-intercept, which is the point where the line crosses the y-axis.

Finding the Equation of a Straight Line

To determine the equation of a straight line, you need the gradient and at least one point on the line. The gradient can be calculated if you have two points on the line.

Calculating the Gradient

The gradient (m) of a line through two points, (x_1, y_1) and (x_2, y_2), is calculated using the formula:

m = (y_2 - y_1) / (x_2 - x_1)

For example, consider the points (2, 3) and (4, 7):

1. Identify the coordinates: (x_1, y_1) = (2, 3) and (x_2, y_2) = (4, 7).
2. Substitute the values into the gradient formula:

m = (7 - 3) / (4 - 2) = 4 / 2 = 2

The gradient of the line is 2.

Using a Point and the Gradient to Find the Equation

Now that we have the gradient, we can use one of the points to find the equation of the line. We will use the point (2, 3).

Substituting the gradient and the point into the equation of a straight line:

y - y_1 = m(x - x_1)

Substituting the known values:

y - 3 = 2(x - 2)

Now, simplify the equation:

y - 3 = 2x - 4

y = 2x - 4 + 3

y = 2x - 1

The equation of the straight line is y = 2x - 1.

Different Forms of the Equation of a Straight Line

There are different forms of the equation of a straight line, including:

• Slope-Intercept Form: y = mx + c
• Point-Slope Form: y - y_1 = m(x - x_1)
• Standard Form: Ax + By + C = 0

Each form is useful in different situations. For example, the slope-intercept form is useful for quickly identifying the gradient and y-intercept, while the standard form is often used in algebraic manipulation.

Example of Converting Between Forms

Consider the equation 2x - 3y + 6 = 0. We can convert it to slope-intercept form:

1. Rearrange the equation to solve for y:

-3y = -2x - 6

y = (2/3)x + 2

The slope-intercept form is y = (2/3)x + 2, where the gradient is 2/3 and the y-intercept is 2.

Horizontal and Vertical Lines

Horizontal lines have a gradient of 0 and can be expressed as:

y = c

where c is the y-coordinate of any point on the line. For example, the line passing through (0, 4) is:

y = 4

Vertical lines have an undefined gradient and can be expressed as:

x = a, where a is the x-coordinate of any point on the line. For example, the line passing through (3, 0) is:

x = 3

Graphing the Equation of a Straight Line

To graph a straight line, you can use the slope-intercept form. Start by plotting the y-intercept (c) on the y-axis. Then, use the gradient (m) to determine another point on the line.

For example, to graph the line y = 2x - 1:

1. Plot the y-intercept at (0, -1).
2. From this point, use the gradient of 2 (rise over run) to find another point. From (0, -1), move up 2 units and right 1 unit to (1, 1).
3. Draw a straight line through the points (0, -1) and (1, 1).

Check Your Understanding

1. Find the equation of the line that passes through the points (1, 2) and (3, 6).

2. Convert the equation 4x - 2y = 8 to slope-intercept form.

3. Identify the gradient and y-intercept of the line represented by the equation y = -3x + 5.

4. Determine the equation of the horizontal line that passes through the point (0, 2).