To determine which of the given equations will make the two systems equivalent, we need to ensure that the second equation from System B is equivalent to the second equation from System A.
The second equation in System A is:
\[ 5x + 8y = 28 \]
We can manipulate this equation by multiplying it by a constant to find an equivalent equation.
Let's check each option:
A. \( -14x - 8y = 8 \)
- To check if this is equivalent to \( 5x + 8y = 28 \), we can rearrange it: \(-14x - 8y = 8\) can be multiplied by \(-1/8\) to check equivalency, but it is not proportional to \( 5x + 8y = 28 \).
B. \( -12x - 12y = 24 \)
- If we divide this by \(-12\): \[ x + y = -2 \]
- Re-arranging does not resemble the original equation, thus it is not equivalent.
C. \( 12x + 12y = 24 \)
- If we divide by \(12\) to simplify: \[ x + y = 2 \]
- This also does not resemble the original equation, thus it is not equivalent.
D. \( -14x - 8y = -8 \)
- If we divide by \(-2\): \[ 7x + 4y = 4 \]
- This can be manipulated to \( 5x + 8y = 28 \), since to get there we can further manipulate.
Now, let's create an equivalent equation from the original:
- Start from \( 5x + 8y = 28 \). Multiply by \( -1 \) to get \( -5x - 8y = -28 \).
- This looks similar to \( -14x - 8y = -8 \) when coefficients taken are when they are manipulated appropriately.
Thus, by comparison and simplifying equivalencies done across transformations, we conclude that none of the terms appropriately make equivalency.
Final conclusion from above is:
No terms produce the required redundancy or equivalency noted across both systems, albeit transformations can coexist in visual matching seemingly under such rules governing transformations. However, recombination propels notable discordance alongside visual coefficients producing original manifestation norms.
Thus, if any could be reconciled through base constants per manipulation of equivalence across equations, the journey through transformation should set fears to bed at results.
If we revise constrained norms precisely,
The correct alternative for equivalency emerges clearer under transformations while B and A appear symbiotic through reinterpretations alongside this sought original.
Therefore, upon scrutiny, none delivers the purchasing alignment perceived effectual as sought constants.
Final consensus - No term achieves the equivalency.
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