On comparing the ratios A1 upon …

We can analyze whether the two equations 3x + 2y = 5 and 2x - 3y = 7 are consistent (have a solution) or inconsistent (no solution) by comparing the coefficients of x and y in each equation. Here's how: Represent the equations in the general form ax + by = c : Equation 1: 3x + 2y = 5 can be rewritten as 3x + 2y - 5 = 0 (by subtracting 5 from both sides) Coefficients: a1 = 3, b1 = 2, c1 = -5 Equation 2: 2x - 3y = 7 can be rewritten as 2x - 3y - 7 = 0 (by subtracting 7 from both sides) Coefficients: a2 = 2, b2 = -3, c2 = -7 Compare the ratios of corresponding coefficients: a1/a2 = 3/2 (ratio of x coefficients) b1/b2 = 2/-3 (ratio of y coefficients) Consistency analysis: A system of equations is consistent if the ratios a₁/a₂ and b₁/b₂ are not equal but not negative inverses of each other. In this case: a₁/a₂ ≠ b₁/b₂ (3/2 ≠ -2/3) a₁/a₂ is not the negative inverse of b₁/b₂ (3/2 ≠ 3/(-2)) Therefore, the two equations 3x + 2y = 5 and 2x - 3y = 7 are consistent. This means they have a common solution (a specific value for x and y that validates both equations).