Here's a step-by-step analysis of the given options and statements. Option (a): The elimination approach on the system x + 3y = 5 and 2x + 6y = 8 - Start by multiplying Eqn (1) by 2 to align the x- and y-terms: 2x + 6y = 10. - Subtract the second equation (2x + 6y = 8) from this result to eliminate x and y: (2x + 6y) - (2x + 6y) = 10 - 8, which yields 0 = 2. - This is a contradiction, which indicates the system has no solution and is inconsistent. - Therefore, the elimination method correctly shows inconsistency for this specific pair of equations. The key misconception to avoid here would be to continue solving as if a unique solution existed; the correct interpretation of 0 = 2 is that no solution satisfies both equations simultaneously. Option (b) (i): Using the inverse matrix method on the system 3x − 5y = 1 and 6x − 10y = 2 - Write the augmented matrix form as [ [3, -5], [6, -10] ] * [ [x], [y] ] = [ [1], [2] ]. - Compute the determinant of the coefficient matrix: det = 3*(-10) - 6*(-5) = -30 + 30 = 0. - A zero determinant means the coefficient matrix is singular and cannot be inverted. Consequently, the inverse matrix method cannot be applied to this system. - This analysis correctly identifies a major limitation: when det = 0, unique solutions via the inverse do not exist, and one must consider other possibilities such as infinitely many solutions or no solution. A common misconception would be to assume an inverse always exists or to ignore the determinant result. Option (b) (ii): Interpreting det = 0 as indicating either infinitely many solutions or no solution; here the equations are multiples of each other - When det = 0, the system may be consistent with dependent equations (infinitely many solutions) or inconsistent (no solution). - To determine which case applies, check if the second equation is a scalar multiple of the first. Here, 6x − 10y = 2 is exactly 2 times (3x − 5y = 1), so the two equations are dependent. - This means the system has infinitely many solutions lying along the line 3x − 5y = 1 (which is equivalent to 6x − 10y = 2). The error to avoid is thinking that det = 0 always yields no solution; in this case, because of the proportional equations, we have infinitely many solutions. Option (b) (iii): Stating the solution set is infinitely many solutions along the line 3x − 5y = 1 - Since the equations are multiples of each other, any solution to 3x − 5y = 1 satisfies both equations, giving an entire line of solutions in the (x, y) plane. - Expressing the solution as all pairs (x, y) such that 3x − 5y = 1 correctly characterizes the solution set. A potential pitfall is mischaracterizing the line or trying to present a single numeric solution when there are infinitely many. In summary, option (a) correctly identifies inconsistency for the first system, option (b) (i) correctly notes that the inverse matrix method cannot be used due to det = 0, option (b) (ii) correctly explains that det = 0 leaves two possibilities and correctly identifies this system as dependent, and option (b) (iii) correctly describes the infinite solution set along the line 3x − 5y = 1. The analysis emphasizes the distinction between inconsistency, dependence, and the nature of the solution set in systems with singular coefficient matrices.

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Here's a step-by-step analysis of the given options and statements. Option (a): The elimination approach on the system x + 3y = 5 and 2x + 6y = 8 - Start by multiplying Eqn (1) by 2 to align the x- and y-terms: 2x + 6y = 10. - Subtract the second equation (2x + 6y = 8) from this result to eliminate x and y: (2x + 6y) - (2x + 6y) = 10 - 8, which yields 0 = 2. - This is a contradiction, which indicates the system has no solution and is inconsistent. - Therefore, the elimination method correctly shows inconsistency for this specific pair of equations. The key misconception to avoid here would be to continue solving as if a unique solution existed; the co......Login to view full explanation

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