Final answer:
When comparing the ratios of the coefficients from the equations 3x + 2y = 5 and 2x - 3y = 7, the ratios a1/a2 and b1/b2 are not equal, indicating that the equations represent lines that intersect at exactly one point. Therefore, the system of equations is consistent and has a unique solution.
Step-by-step explanation:
To determine whether the pair of linear equations 3x + 2y = 5 and 2x - 3y = 7 are consistent or inconsistent, we compare the ratios of the coefficients of x and y, and the constants in both equations. The method we use is based on the comparison of ratios a₁/a₂, b₁/b₂, and c₁/c₂ where a₁, b₁ are the coefficients of x and y in the first equation, a₂, b₂ are the coefficients of x and y in the second equation, and c₁, c₂ are the constants.
- For the first equation (3x + 2y = 5), the coefficients are a₁ = 3 and b₁ = 2. The constant is c₁ = 5.
- For the second equation (2x - 3y = 7), the coefficients are a₂ = 2 and b₂ = -3. The constant is c₂ = 7.
- Compare the ratios: a₁/a₂ = 3/2, b₁/b₂ = 2/(-3), c₁/c₂ = 5/7.
- Since the ratios a₁/a₂ and b₁/b₂ are not equal, the lines represented by these equations are not parallel, and they intersect at exactly one point. This means that the system of equations is consistent and has a unique solution.