Linear Graphs in Mathematics: Visualizing Linear Relationships
Learn about linear graphs, visual representations of linear equations. This guide explains how to create and interpret linear graphs, highlighting their use in visualizing linear relationships between variables and contrasting them with line graphs.
Linear Graphs in Discrete Mathematics
What is a Linear Graph?
A linear graph is a visual representation of a linear equation. A linear equation is an equation whose graph is a straight line. Linear graphs show the relationship between two variables where a constant change in one variable results in a proportional change in the other.
Example: A Linear Graph
Imagine Harry earns $15 per hour and has daily expenses of $50. To figure out how many hours he needs to work to have savings, we can use a linear graph. His income (l) is given by the linear equation l = 15t, where t is the number of hours worked.
(An illustrative graph of l = 15t would be included here, showing how the income increases linearly with the hours worked. The graph should also indicate Harry's daily expenses and show a region where his income exceeds his expenses.)
Linear Graphs vs. Line Graphs
While both involve lines, there's a key difference: A linear graph shows a single straight line representing a linear relationship, whereas a line graph can show multiple line segments that may or may not form a straight line. The points in a linear graph are always collinear (lie on a single line).
(An illustrative comparison of a linear graph and a line graph would be included here, highlighting the difference in the arrangement of points and lines.)
Equation of a Linear Graph
The general equation for a linear graph is Ax + By = C, where A, B, and C are real numbers, and A and B are not both zero. For example, 2x + 5y = 0 is a linear equation.
Plotting a Linear Equation
- Identify the two variables (e.g., x and y).
- Find three points (x, y) that satisfy the equation (to ensure accuracy).
- Create a table of these points.
- Plot the points on a Cartesian plane.
- Draw a straight line through the points, extending it in both directions.
(A worked example showing how to plot a linear equation, such as 2x + y = 6, on a graph would be included here, including the table of points and the graph itself.)
Important Considerations When Graphing
- While two points define a line, plotting a third point is a good way to verify the accuracy of your work.
- A horizontal line is parallel to the x-axis and has the equation y = kx (where k is a constant).
- A vertical line is parallel to the y-axis and has the equation x = ky (where k is a constant).
Conclusion
Linear graphs are a fundamental tool for visually representing linear relationships between variables. They provide a simple yet powerful way to understand and analyze linear equations.