1. Understanding the Intercept Form
The intercept form of the equation of a line is used when we know where the line cuts the x-axis and y-axis. The points where the line intersects these axes are called the x-intercept and y-intercept.
The intercept form gives a simple and clear way to write the equation using these two values.
1.1. Definition
If a line cuts the x-axis at \(a\) and the y-axis at \(b\), then the equation of the line in intercept form is:
\( \dfrac{x}{a} + \dfrac{y}{b} = 1 \)
2. Understanding x-intercept and y-intercept
The intercepts represent the points where the line meets the axes.
2.1. x-intercept (a)
The x-intercept is the value of x where the line crosses the x-axis. At this point, \(y = 0\). So the point is \((a, 0)\).
2.2. y-intercept (b)
The y-intercept is the value of y where the line crosses the y-axis. At this point, \(x = 0\). So the point is \((0, b)\).
2.2.1. Visual Understanding
The intercept form shows how the line stretches between these two intercept points on the axes.
3. How to Use the Intercept Form
To write the equation of a line in intercept form, simply substitute the x-intercept and y-intercept values into the formula.
3.1. Steps
- Identify the x-intercept \(a\).
- Identify the y-intercept \(b\).
- Write the equation as
\( \dfrac{x}{a} + \dfrac{y}{b} = 1 \)
4. Examples
The following examples show how to form equations using intercepts.
4.1. Example 1: Both Intercepts Positive
A line cuts the x-axis at 4 and the y-axis at 6.
\( \dfrac{x}{4} + \dfrac{y}{6} = 1 \)
4.2. Example 2: One Negative Intercept
x-intercept = -5, y-intercept = 3.
\( \dfrac{x}{-5} + \dfrac{y}{3} = 1 \)
4.3. Example 3: Vertical and Horizontal Considerations
If one of the intercepts is zero, the intercept form cannot be used directly.
- If x-intercept = 0 → the line is vertical and the equation is \(x = 0\).
- If y-intercept = 0 → the line passes through the origin and equation becomes \(y = mx\) using slope-intercept form.
4.4. Example 4: Converting to Slope-Intercept Form
Given:
\( \dfrac{x}{2} + \dfrac{y}{3} = 1 \)
Rewrite:
\( y = 3 - \dfrac{3x}{2} \)
So slope-intercept form is:
\( y = -\dfrac{3}{2}x + 3 \)