Question

Solve the following system of equations graphically : \(x + 3y = 6\) and \(2x - 3y = 12\). Also, find the area of the triangle formed by the lines \(x + 3y = 6\), \(x = 0\) and \(y = 0\).

Hint:

"Supplementary" means sum to 180°, while "Complementary" means sum to 90°. A quick way to remember: 'C' comes before 'S', and 90 comes before 180.

Answer 1

Step 1: Understanding the Concept:
Graphical solution involves plotting both lines on a coordinate plane and finding their point of intersection. The triangle formed by \(x=0\) (y-axis), \(y=0\) (x-axis), and a line is a right-angled triangle.
Step 2: Key Formula or Approach:
1. Find intercepts for both lines. 2. Area of triangle = \(\frac{1}{2} \times \text{base} \times \text{height}\).
Step 3: Detailed Explanation:
1. For \(x + 3y = 6\): If \(x=0, y=2\); If \(y=0, x=6\). Points: \((0, 2), (6, 0)\).
2. For \(2x - 3y = 12\): If \(x=0, y=-4\); If \(y=0, x=6\). Points: \((0, -4), (6, 0)\).
3. Intersection: Both lines pass through \((6, 0)\). So, the solution is \(x = 6, y = 0\).
4. Triangle Area: The lines \(x+3y=6, x=0, y=0\) form a triangle with vertices \((0,0), (6,0), \text{ and } (0,2)\).
- Base (along x-axis) = 6 units.
- Height (along y-axis) = 2 units.
- Area = \(\frac{1}{2} \times 6 \times 2 = 6\) sq units.
Step 4: Final Answer:
Solution: \(x=6, y=0\). Area of the triangle is 6 sq units.