Understanding the equations of straight-line graphs is foundational in coordinate geometry. This section delves into interpreting and formulating linear equations, emphasizing various forms such as and , focusing on identifying the gradient (slope) and y-intercept.
![Linear function graph](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2Fa54ff0d5-33e9-4d64-8130-6b149ed7b466-file.png&w=1200&q=75)
Image courtesy of HMH
What is a Linear Equation?
A linear equation represents a straight line on a coordinate plane, typically expressed in the form , where:
- is the gradient, determining the steepness and direction of the line.
- is the y-intercept, indicating where the line crosses the y-axis.
Another common form is , which can be rearranged into or other required forms.
Interpreting Straight-Line Graphs
To interpret straight-line graphs, one must grasp two main components: the gradient and the y-intercept.
- Gradient (m): Shows the change in for a unit change in . Positive gradients indicate an upward slope, negative gradients a downward slope, and a zero gradient indicates a horizontal line.
- Y-intercept (c): The point where the line intersects the y-axis, determined by the value when .
Formulating Linear Equations
Creating equations of straight lines involves finding the gradient and y-intercept from a graph or set of points.
From a Graph
Directly identify the y-intercept where the line crosses the y-axis. To find the gradient, select two points on the line, applying:
![Slope](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2Fe15bdd2b-3390-4a86-9e55-98e9967cd35f-file.png&w=1080&q=75)
From Points
Given two points and , calculate the gradient using the same formula for . Once is known, use with one point to solve for .
![Slope illustration](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2F50e73296-4c91-49ae-90aa-43fd9dcdb7f9-file.png&w=1080&q=75)
Image courtesy of CueMath
Worked Examples
Example 1: Finding the Equation from a Gradient and a Point
Given a gradient and a point , find the equation of the line.
Solution:
1. Using the formula , we substitute and the point to solve for :
2. Solving for , we find .
Thus, the equation of the line is:
Example 2: Determining Gradient and Y-intercept from an Equation
Given the equation , rearrange it to the form to identify the gradient and y-intercept.
Solution:
1. Rearrange the equation:
2. Gradient ():
3. Y-intercept ():