(05.03 MC) Determine which system will produce infinitely many solutions. \( \begin{array}{l}2 x+5 y=24 \\ 2 x+5 y=42\end{array} \) \( \begin{array}{l}\text { Ox } \\ 6 x+5 y=11 \\ -8 x+6 y=-18\end{array} \) \( \begin{array}{l}\text { O } 2 x-3 y=9 \\ -2 x+3 y=-7\end{array} \)

To determine which system of equations will produce infinitely many solutions, we need to check if the two equations in each system are identical or if they represent the same line. Let's analyze each system of equations: 1. \( \begin{array}{l}2x + 5y = 24 \\ 2x + 5y = 42\end{array} \) In this system, both equations are identical, which means they represent the same line. Therefore, this system will produce infinitely many solutions. 2. \( \begin{array}{l}6x + 5y = 11 \\ -8x + 6y = -18\end{array} \) To check if these equations represent the same line, we can multiply the first equation by 2 and the second equation by 3 to make the coefficients of \(x\) and \(y\) the same: \( 12x + 10y = 22 \) \( -24x + 18y = -54 \) Now, we can see that the coefficients of \(x\) and \(y\) are the same, but the constants are different. This means that these equations do not represent the same line, and therefore, this system will not produce infinitely many solutions. 3. \( \begin{array}{l}2x - 3y = 9 \\ -2x + 3y = -7\end{array} \) To check if these equations represent the same line, we can add the two equations together to eliminate the variable \(x\): \( (2x - 3y) + (-2x + 3y) = 9 + (-7) \) \( 0 = 2 \) This result is a contradiction, which means that these equations do not represent the same line. Therefore, this system will not produce infinitely many solutions. In conclusion, the first system of equations will produce infinitely many solutions.