Check graphically whether the pair of equations x + 3y = 6 (1) and 2x – 3y = 12 (2) is consistent. If so, solve them graphically.
Step-by-Step Solution
Key Concept: A pair of linear equations is *consistent* if their graphs intersect at a point. The coordinates of the intersection give the solution of the system. To check graphically we plot each line using two convenient points, draw the lines, and observe whether they meet.
1. Rewrite each equation in intercept form
- For (1) : $x + 3y = 6 \Rightarrow 3y = 6 - x \Rightarrow y = 2 - \frac{x}{3}$
- For (2) : $2x - 3y = 12 \Rightarrow -3y = 12 - 2x \Rightarrow y = \frac{2x - 12}{3}$
2. Find two points on each line
- Line (1)
*When $x = 0$*: $3y = 6 \Rightarrow y = 2$ → point $A(0,2)$
*When $y = 0$*: $x = 6$ → point $B(6,0)$
- Line (2)
*When $x = 0$*: $-3y = 12 \Rightarrow y = -4$ → point $C(0,-4)$
*When $y = 0$*: $2x = 12 \Rightarrow x = 6$ → point $D(6,0)$
3. Plot the points on the Cartesian plane
- Mark $A(0,2)$ and $B(6,0)$ and join them – this is the graph of equation (1).
- Mark $C(0,-4)$ and $D(6,0)$ and join them – this is the graph of equation (2).
4. Observe the intersection
- Both lines pass through the common point $B(6,0)$. Hence the two lines intersect at a single point.
5. Conclusion about consistency
- Since the lines intersect, the pair of equations is *consistent* (they have a unique solution).
6. Solution (graphical coordinates of intersection)
- The intersection point is $\boxed{(6,\,0)}$.
7. Verification (optional algebraic check)
- Substituting $x=6, y=0$ in (1): $6 + 3\times0 = 6$ ✓
- Substituting in (2): $2\times6 - 3\times0 = 12$ ✓
- Hence the graphical solution matches the algebraic solution.
Correct Answer: The pair is consistent. Their graphical intersection gives the unique solution $x = 6$, $y = 0$ (i.e., the point (6, 0)).