Question

The system of equations $x + 2y = 3$ and $2x + 3y = 3$ has

• A. No solution
• B. Unique solution
• C. Infinite solutions
• D. Only two solutions

Hint:

Always check the ratio of the $x$ and $y$ coefficients first. If they are different, you immediately know it's a unique solution without even looking at the constant terms ($c_1, c_2$).

Answer 1

The Correct Option is B

To determine the nature of the solution for the given system of equations, we use the method of elimination or substitution to find if there is a unique solution, no solution, or infinite solutions.

Given System of Equations:

• \(x + 2y = 3\)
• \(2x + 3y = 3\)

Step-by-step Solution:

1. Elimination Method: We will use the elimination method to solve the system of equations by eliminating one of the variables.
   1. Multiply the first equation by 2 to align the coefficients of \(x\):
      • \(2(x + 2y) = 2 \cdot 3 \Rightarrow 2x + 4y = 6\)
   2. Subtract the second equation from the modified first equation:
      • \((2x + 4y) - (2x + 3y) = 6 - 3\)
      • \(2x + 4y - 2x - 3y = 3\)
      • \(y = 3\)
   3. Substitute \(y = 3\) back into the first equation to find \(x\):
      • \(x + 2(3) = 3\)
      • \(x + 6 = 3\)
      • \(x = 3 - 6 = -3\)
2. The solution is \((x, y) = (-3, 3)\).

Conclusion:

The system of equations has a unique solution: \(x = -3\) and \(y = 3\), hence the correct answer is Unique solution.