Question

The pair of linear equations \( 2x - 3y = 8 \) and \( 4x - 6y = 9 \) represents the following:

• A. The system has a unique solution.
• B. The system has infinitely many solutions.
• C. The system has no solution.
• D. The system represents two parallel lines.

Hint:

If the two equations are multiples of each other but have different constants, the system has no solution (the lines are parallel).

Answer 1

The Correct Option is D

Step 1: Write the equations.
The given equations are: \[ 2x - 3y = 8 \quad \text{(Equation 1)} \] \[ 4x - 6y = 9 \quad \text{(Equation 2)}. \] Step 2: Observe the relationship between the two equations.
Notice that Equation 2 is exactly twice Equation 1: \[ 4x - 6y = 2(2x - 3y) = 16. \] But Equation 2 is given as \( 4x - 6y = 9 \), not 16. Step 3: Conclude the system's nature.
Since the left-hand sides are proportional, but the right-hand sides are not, the system represents two parallel lines. Parallel lines do not intersect, so there is no solution.