Graphical Solution, Shaded Region and Area
Video Explanation
Question
Solve the following system of linear equations graphically. Shade the region bounded by these lines and the y-axis. Also find the area of the bounded region:
\[ 3x + y – 11 = 0 \]
\[ x – y – 1 = 0 \]
Solution
Step 1: Write Both Equations in the Form \(y = mx + c\)
Equation (1):
\[ 3x + y – 11 = 0 \Rightarrow y = 11 – 3x \]
Equation (2):
\[ x – y – 1 = 0 \Rightarrow y = x – 1 \]
Step 2: Prepare Tables of Values
For Equation (1): \(y = 11 – 3x\)
| x | y |
|---|---|
| 0 | 11 |
| 3 | 2 |
For Equation (2): \(y = x – 1\)
| x | y |
|---|---|
| 0 | -1 |
| 3 | 2 |
Step 3: Graphical Representation
Plot the following points on the same Cartesian plane:
- Line 1: (0, 11) and (3, 2)
- Line 2: (0, −1) and (3, 2)
Join each pair of points to obtain two straight lines.
The two straight lines intersect at the point (3, 2).
Step 4: Region Bounded with the Y-Axis
The bounded region is formed by:
- Line \(3x + y – 11 = 0\)
- Line \(x – y – 1 = 0\)
- The y-axis \((x = 0)\)
Shade the triangular region enclosed by these two lines and the y-axis.
Step 5: Area of the Bounded Region
Vertices of the triangle are:
- (0, 11)
- (0, −1)
- (3, 2)
Base of the triangle (along y-axis) = \(11 – (-1) = 12\) units
Height of the triangle = horizontal distance of point (3, 2) from y-axis = 3 units
\[ \text{Area} = \frac{1}{2} \times 12 \times 3 = 18 \]
Answer
The graphical solution of the given system of equations is:
\[ (x, y) = (3, 2) \]
Area of the region bounded by the given lines and the y-axis = 18 square units.
Conclusion
The required triangular region is shaded and its area is 18 square units.