Graphical Solution and Shading of the Required Region
Video Explanation
Question
Solve the following system of linear equations graphically and shade the region between the two lines and the x-axis:
\[ 2x + 3y = 12 \]
\[ x – y = 1 \]
Solution
Step 1: Write Both Equations in the Form \(y = mx + c\)
Equation (1):
\[ 2x + 3y = 12 \Rightarrow 3y = 12 – 2x \Rightarrow y = 4 – \frac{2}{3}x \]
Equation (2):
\[ x – y = 1 \Rightarrow y = x – 1 \]
Step 2: Prepare Tables of Values
For Equation (1): \(y = 4 – \frac{2}{3}x\)
| x | y |
|---|---|
| 0 | 4 |
| 6 | 0 |
For Equation (2): \(y = x – 1\)
| x | y |
|---|---|
| 1 | 0 |
| 3 | 2 |
Step 3: Graphical Representation
Plot the following points on the same Cartesian plane:
- Line 1: (0, 4) and (6, 0)
- Line 2: (1, 0) and (3, 2)
Join each pair of points to obtain two straight lines.
The two straight lines intersect at the point (3, 2).
Result
The graphical solution of the given system of equations is:
\[ x = 3,\quad y = 2 \]
Step 4: Shading of the Required Region
The required region is the region enclosed by:
- The line \(2x + 3y = 12\)
- The line \(x – y = 1\)
- The x-axis \((y = 0)\)
Shade the triangular region formed between these two lines and the x-axis.
Conclusion
The given system of linear equations has a unique solution at the point (3, 2).
The shaded region represents the region bounded by the two given lines and the x-axis.