Question

The pair of equations \( x + 3y = 6 \) and \( 2x - 3y = 12 \) :

• A. is consistent
• B. is not consistent
• C. has a unique solution
• D. does not have an infinite solution

Hint:

For a system \( a_1x + b_1y = c_1 \) and \( a_2x + b_2y = c_2 \): - Unique solution if \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \) (Consistent). - Infinite solutions if \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \) (Consistent). - No solution if \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \) (Inconsistent).

Answer 1

The Correct Option is C

Step 1: Consider the standard form of a pair of linear equations in two variables: \[ a_1x + b_1y = c_1 \] \[ a_2x + b_2y = c_2 \] For the given equations: Equation 1: \( x + 3y = 6 \implies a_1 = 1, b_1 = 3, c_1 = 6 \) Equation 2: \( 2x - 3y = 12 \implies a_2 = 2, b_2 = -3, c_2 = 12 \) Step 2: Check the conditions for consistency and the type of solution by comparing the ratios of the coefficients. Calculate the ratios: \[ \frac{a_1}{a_2} = \frac{1}{2} \] \[ \frac{b_1}{b_2} = \frac{3}{-3} = -1 \] \[ \frac{c_1}{c_2} = \frac{6}{12} = \frac{1}{2} \] Step 3: Analyze the ratios. We compare \( \frac{a_1}{a_2} \) and \( \frac{b_1}{b_2} \). \[ \frac{1}{2} \neq -1 \] Since \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \), the system of equations has a unique solution. Step 4: Determine the consistency based on the type of solution. A system with a unique solution is always consistent. Step 5: Evaluate the options based on the findings. % Option (A) "is consistent": True, because it has a unique solution. % Option (B) "is not consistent": False. % Option (C) "has a unique solution": True. This is the specific condition we found. % Option (D) "does not have an infinite solution": True, because it has a unique solution. Step 6: Select the most appropriate option. While options (A), (C), and (D) are all technically correct statements about this system, option (C) provides the most specific and complete description of the nature of the solution based on the coefficient ratios. A system with a unique solution is inherently consistent and does not have infinite solutions. Therefore, "has a unique solution" is the best description among the choices.