Graphical Solution, Shaded Region and Area
Video Explanation
Question
Solve the following system of equations graphically and shade the region between the lines and the y-axis:
\[ 4x – y = 4 \]
\[ 3x + 2y = 14 \]
Solution
Step 1: Write Both Equations in the Form \(y = mx + c\)
Equation (1):
\[ 4x – y = 4 \Rightarrow y = 4x – 4 \]
Equation (2):
\[ 3x + 2y = 14 \Rightarrow 2y = 14 – 3x \Rightarrow y = 7 – \frac{3}{2}x \]
Step 2: Prepare Tables of Values
For Equation (1): \(y = 4x – 4\)
| x | y |
|---|---|
| 0 | -4 |
| 1 | 0 |
For Equation (2): \(y = 7 – \frac{3}{2}x\)
| x | y |
|---|---|
| 0 | 7 |
| 2 | 4 |
Step 3: Graphical Representation
Plot the following points on the same Cartesian plane:
- Line 1: (0, −4) and (1, 0)
- Line 2: (0, 7) and (2, 4)
Join each pair of points to obtain two straight lines.
The two straight lines intersect at the point (2, 4).
Result
The graphical solution of the given system of equations is:
\[ (x, y) = (2, 4) \]
Step 4: Shading of the Required Region
The region bounded by:
- The line \(4x – y = 4\)
- The line \(3x + 2y = 14\)
- The y-axis \((x = 0)\)
forms a triangular region. Shade this region.
Step 5: Area of the Bounded Region
Vertices of the triangle are:
- (0, −4)
- (0, 7)
- (2, 4)
Base of the triangle (along y-axis) = \(7 – (-4) = 11\) units
Height of the triangle = horizontal distance of (2, 4) from y-axis = 2 units
\[ \text{Area} = \frac{1}{2} \times 11 \times 2 = 11 \]
Answer
Solution of the given system of equations is:
\[ (x, y) = (2, 4) \]
Area of the region bounded by the given lines and the y-axis = 11 square units.
Conclusion
The required region is shaded and its area is 11 square units.