Graphical Solution and Area of the Region Bounded by Lines and X-Axis
Video Explanation
Question
Solve graphically the following system of linear equations and find the area bounded by these lines and the x-axis:
\[ 4x – 3y + 4 = 0 \]
\[ 4x + 3y – 20 = 0 \]
Solution
Step 1: Write Both Equations in the Form \(y = mx + c\)
Equation (1):
\[ 4x – 3y + 4 = 0 \Rightarrow -3y = -4x – 4 \Rightarrow y = \frac{4}{3}x + \frac{4}{3} \]
Equation (2):
\[ 4x + 3y – 20 = 0 \Rightarrow 3y = 20 – 4x \Rightarrow y = \frac{20}{3} – \frac{4}{3}x \]
Step 2: Prepare Tables of Values
For Equation (1): \(y = \frac{4}{3}x + \frac{4}{3}\)
| x | y |
|---|---|
| -1 | 0 |
| 2 | 4 |
For Equation (2): \(y = \frac{20}{3} – \frac{4}{3}x\)
| x | y |
|---|---|
| 5 | 0 |
| 2 | 4 |
Step 3: Graphical Representation
Plot the following points on the same Cartesian plane:
- Line 1: (−1, 0) and (2, 4)
- Line 2: (5, 0) and (2, 4)
Join each pair of points to obtain two straight lines.
The two straight lines intersect at the point (2, 4).
Step 4: Triangle Formed with the X-Axis
The triangle is formed by:
- Intersection of \(4x – 3y + 4 = 0\) with x-axis → (−1, 0)
- Intersection of \(4x + 3y – 20 = 0\) with x-axis → (5, 0)
- Intersection point of the two lines → (2, 4)
Step 5: Area of the Triangle
Base of the triangle = distance between (−1, 0) and (5, 0) = 6 units
Height of the triangle = y-coordinate of the vertex (2, 4) = 4 units
\[ \text{Area} = \frac{1}{2} \times 6 \times 4 = 12 \]
Answer
The graphical solution of the given system of equations is:
\[ (x, y) = (2, 4) \]
Area bounded by the given lines and the x-axis = 12 square units.
Conclusion
The triangle formed by the two given lines and the x-axis is shaded and its area is 12 square units.