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

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

Step 1: Standard Form
Equation (1): \( x + 3y = 6 \) (already standard)
Equation (2): Rewrite \( 3y - 2x = -12 \):
\[ -2x + 3y = -12 \Rightarrow 2x - 3y = 12 \]
Resulting system:
\[ (1)\ x + 3y = 6\\\ (2)\ 2x - 3y = 12 \]

Step 2: Eliminate \( y \)
Add equations (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: System Nature
Unique solution: The system is consistent with a unique solution.

Step 4: Graphical Solution
Plot the equations to find the intersection point.
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)
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 \( (6, 0 ) \).

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