To determine which statement is correct, we can analyze the provided systems of linear equations. We will check for consistency (no solution, unique solution, infinite solutions) in each pair.
-
6x - 5y = 8 and 12x - 10y = 16
The second equation can be expressed as: \[ 12x - 10y = 2 \cdot (6x - 5y) = 2 \cdot 8 = 16 \] Since the second equation is a multiple of the first one, both equations represent the same line. Therefore, they have an infinite number of solutions.
-
7x + 2y = 6 and 14x + 4y = 16
The second equation can be expressed as: \[ 14x + 4y = 2 \cdot (7x + 2y) = 2 \cdot 6 = 12 \] Since 12 does not equal 16, these equations are inconsistent and have no solution.
-
8x - 3y = 10 and 16x - 6y = 22
The second equation can be expressed as: \[ 16x - 6y = 2 \cdot (8x - 3y) = 2 \cdot 10 = 20 \] Since 20 does not equal 22, these equations are inconsistent and have no solution.
-
9x + 6y = 14 and 18x + 12y = 26
The second equation can be expressed as: \[ 18x + 12y = 2 \cdot (9x + 6y) = 2 \cdot 14 = 28 \] Since 28 does not equal 26, these equations are inconsistent and have no solution.
Based on this analysis, here is the summary of the statements:
- The first statement is incorrect (infinite solutions).
- The second statement is correct (no solution).
- The third statement is correct (no solution).
- The fourth statement is incorrect (no solution).
Thus, the correct statement is:
The system of linear equations 7 x + 2 y = 6 and 14 x + 4 y = 16 has no solution.
1 answer