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:
\[ 3x + 2y – 4 = 0 \]
\[ 2x – 3y – 7 = 0 \]
Solution
Step 1: Write Both Equations in the Form \(y = mx + c\)
Equation (1):
\[ 3x + 2y – 4 = 0 \Rightarrow 2y = 4 – 3x \Rightarrow y = 2 – \frac{3}{2}x \]
Equation (2):
\[ 2x – 3y – 7 = 0 \Rightarrow -3y = 7 – 2x \Rightarrow y = \frac{2}{3}x – \frac{7}{3} \]
Step 2: Prepare Tables of Values
For Equation (1): \(y = 2 – \frac{3}{2}x\)
| x | y |
|---|---|
| 0 | 2 |
| \(\frac{4}{3}\) | 0 |
For Equation (2): \(y = \frac{2}{3}x – \frac{7}{3}\)
| x | y |
|---|---|
| 0 | \(-\frac{7}{3}\) |
| \(\frac{7}{2}\) | 0 |
Step 3: Graphical Representation
Plot the following points on the same Cartesian plane:
- Line 1: (0, 2) and (4/3, 0)
- Line 2: (0, −7/3) and (7/2, 0)
Join each pair of points to obtain two straight lines.
The two straight lines intersect at the point (2, −1).
Result
The graphical solution of the given system of equations is:
\[ x = 2,\quad y = -1 \]
Step 4: Shading of the Required Region
The required region is the region enclosed by:
- The line \(3x + 2y – 4 = 0\)
- The line \(2x – 3y – 7 = 0\)
- 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 (2, −1).
The shaded region represents the region bounded by the two given lines and the x-axis.