1. Introduction
The graphical method is a visual way to solve linear equations in two variables. Each linear equation represents a straight line on the coordinate plane. The solution of a pair of linear equations is the point where their lines intersect.
This method helps students understand the concept of solutions more clearly—especially how two lines can intersect, be parallel, or overlap.
2. Plotting a Linear Equation on a Graph
To draw the graph of a linear equation, we find two or more solutions (points) and plot them on the coordinate plane. Joining these points gives a straight line.
2.1. Creating a Value Table
Select values for one variable and calculate the other variable. This gives the coordinates of points on the line.
Example: Plot \(2x + y = 6\).
| \(x\) | \(y\) |
|---|---|
| 0 | 6 |
| 1 | 4 |
| 2 | 2 |
Points: \((0,6), (1,4), (2,2)\)
2.2. Plotting Points
Mark each point on the graph using the \(x\)-coordinate (horizontal axis) and \(y\)-coordinate (vertical axis).
After marking all points, draw a straight line through them.
3. Solving Two Linear Equations Graphically
To solve a pair of linear equations using graphs, plot both lines on the same graph. The point of intersection represents the solution \((x, y)\).
3.1. Case 1: One Solution (Intersecting Lines)
If the two lines intersect at exactly one point, that point gives the unique solution.
Example: Lines intersect at \((2,1)\). So the solution is \(x = 2, y = 1\).
3.2. Case 2: No Solution (Parallel Lines)
If the lines do not meet at any point, they are parallel. Parallel lines never intersect, so there is no solution.
In this case, the equations are said to be inconsistent.
3.3. Case 3: Infinite Solutions (Coincident Lines)
If both lines lie on top of each other, they are coincident lines. Every point on one line is also on the other.
So, there are infinitely many solutions and the equations are dependent.
4. Worked Examples
Here are step-by-step examples.
4.1. Example 1
Solve graphically:
\(x + y = 6\)
\(x - y = 2\)
Step 1: Make value table for each equation.
| Equation | \(x\) | \(y\) |
|---|---|---|
| \(x + y = 6\) | 0 | 6 |
| 2 | 4 | |
| \(x - y = 2\) | 0 | -2 |
| 2 | 0 |
Step 2: Plot all points and draw lines.
Step 3: Lines intersect at \((4, 2)\).
Solution: \(x = 4, y = 2\)
4.2. Example 2
Solve graphically:
\(2x + y = 4\)
\(4x + 2y = 8\)
Both equations represent the same line (multiplying the first by 2 gives the second).
Result: Coincident lines → infinitely many solutions.
5. Advantages and Limitations of Graphical Method
- Helps understand visually how solutions work.
- Useful for estimating solutions.
- Not very accurate when dealing with fractional values.
- Graphs may be difficult for large or complex numbers.
6. Quick Practice
Use the graphical method to solve:
- \(x + 2y = 8\) and \(x - y = 1\)
- \(2x - y = 4\) and \(x + y = 3\)
- \(3x + 2y = 12\) and \(6x + 4y = 24\)
7. Summary
- A linear equation in two variables represents a straight line.
- Graphical method solves equations by finding the intersection point.
- Intersecting lines → one solution.
- Parallel lines → no solution.
- Coincident lines → infinitely many solutions.
- Value tables help generate points to plot lines.