Question

Check whether the following pair of equations is consistent or not. If consistent, solve graphically:
\[ x + 3y = 6\\ 3y - 2x = -12 \]

Hint:

If two equations simplify to the same line, the system has infinite solutions.

Answer 1

Given System of Equations:
\[ (1)\ x + 3y = 6\\ (2)\ 3y - 2x = -12 \]

Step 1: Rewrite both equations in standard form
Equation (1): \( x + 3y = 6 \) → already in standard form.
Equation (2): Rearranging \( 3y - 2x = -12 \) to standard form:
\[ -2x + 3y = -12 \Rightarrow 2x - 3y = 12 \]
So, the pair becomes:
\[ (1)\ x + 3y = 6\\ (2)\ 2x - 3y = 12 \]

Step 2: Add the two equations to eliminate \( y \)
Add (1) and (2): \[ x + 3y + 2x - 3y = 6 + 12\\ \Rightarrow 3x = 18 \Rightarrow x = 6 \]
Substitute \( x = 6 \) into equation (1): \[ 6 + 3y = 6 \Rightarrow 3y = 0 \Rightarrow y = 0 \]
Step 3: Nature of the system
We obtained a unique solution, so the system of equations is consistent and has a unique solution.

Step 4: Graphical Solution
To solve graphically, plot both equations and find their point of intersection.
For Equation (1): \( x + 3y = 6 \)
Find two points: - If \( x = 0 \), \( 3y = 6 \Rightarrow y = 2 \) → (0, 2) - If \( y = 0 \), \( x = 6 \) → (6, 0)
For Equation (2): \( 2x - 3y = 12 \)
Find two points: - If \( x = 0 \), \( -3y = 12 \Rightarrow y = -4 \) → (0, -4) - If \( y = 0 \), \( 2x = 12 \Rightarrow x = 6 \) → (6, 0)
Both lines intersect at point \( (6, 0) \).

Final Answer:
The pair of equations is consistent and has a unique solution.
Solution: \( \boxed{x = 6,\ y = 0} \)