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

Step 1: Understanding the Concept:
The nature of solutions for a system of two linear equations in two variables ($a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$) can be determined by comparing the ratios of their coefficients.

Step 2: Key Formula or Approach:
Calculate the ratios $\frac{a_1}{a_2}$ and $\frac{b_1}{b_2}$. - If $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$, the system has a unique solution (intersecting lines). - If $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, the system has infinite solutions (coincident lines). - If $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$, the system has no solution (parallel lines).

Step 3: Detailed Explanation:
The given system of equations is: 1) $1x + 2y = 3$ 2) $2x + 3y = 3$ Here, the coefficients are: $a_1 = 1$, $b_1 = 2$, $c_1 = 3$ $a_2 = 2$, $b_2 = 3$, $c_2 = 3$ Now, find the ratios: $\frac{a_1}{a_2} = \frac{1}{2}$ $\frac{b_1}{b_2} = \frac{2}{3}$ Comparing the ratios: $\frac{1}{2} \neq \frac{2}{3}$ Since $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$, the lines are not parallel and not coincident; they intersect at exactly one point. Therefore, the system has a unique solution.

Step 4: Final Answer:
The system has a unique solution.