realize. \( 3(A-E)-A L G E B R A I-D A V I D S O N-Y R(L M S) \) MIT \( 2 / 3 \) Topic 4 Test (LMS graded) For the system of equations below, select all the equivalent equations for \( 3 x+2 y=16 \) that when used with \( 5 x-4 y=-10 \) will have the same solution. \[ \begin{array}{l} 3 x+2 y=16 \\ 5 x-4 y=-10 \end{array} \] A. \( 6 x+4 y=32 \) B. \( 2 x+y=15 \) C. \( x+\frac{2}{3} y=\frac{16}{3} \) D. \( -12 x-8 y=-64 \) E. \( -12 x+8 y=-64 \) F. \( -6 x+4 y=-32 \)

To find the equivalent equations for \( 3x + 2y = 16 \) that will maintain the same solution set when combined with \( 5x - 4y = -10 \), we can start by manipulating the first equation. A. \( 6x + 4y = 32 \): This is valid since it is just \( 2 \) times the original equation, thus equivalent. B. \( 2x + y = 15 \): This is not equivalent since it does not represent the same linear relationship as the original. C. \( x + \frac{2}{3}y = \frac{16}{3} \): This is equivalent as it can be obtained through scaling down the original equation. D. \( -12x - 8y = -64 \): This is equivalent since it is \( -4 \) multiplied by the original equation. E. \( -12x + 8y = -64 \): This is not equivalent as it wouldn't satisfy the same relationship established by the original. F. \( -6x + 4y = -32 \): This is not equivalent since it cannot be derived from the original equation. So, the correct answers are A, C, and D.