Graph of a Linear Equation

1. Graphical Representation

A linear equation in two variables ($ax + by + c = 0$) represents a straight line on the Cartesian plane. Every point on this line is a solution to the equation, and every solution to the equation is a point on this line.

2. Steps to Draw the Graph

To draw the graph of a linear equation, follow these steps:

1. Express $y$ in terms of $x$ (or $x$ in terms of $y$). e.g., $y = 3x - 2$.
2. Choose at least two convenient values for $x$ and find the corresponding values of $y$.
3. Prepare a table of these solutions $(x, y)$.
4. Plot these points on a graph paper.
5. Join the points with a straight line and extend it in both directions.

Example: Graph of $x + y = 4$

• If $x = 0$, then $0 + y = 4 \Rightarrow y = 4$. Point: $(0, 4)$
• If $x = 4$, then $4 + y = 4 \Rightarrow y = 0$. Point: $(4, 0)$
• If $x = 2$, then $2 + y = 4 \Rightarrow y = 2$. Point: $(2, 2)$

Plotting $(0, 4), (4, 0), (2, 2)$ and joining them gives the graph.

3. Equations of Lines Parallel to Axes

• Equation of X-axis: $y = 0$
• Equation of Y-axis: $x = 0$
• Line parallel to X-axis: $y = k$ (where $k$ is a constant).
• Line parallel to Y-axis: $x = k$ (where $k$ is a constant).