To determine which of the given options will make the systems equivalent, we want to check if the second equation (5x + 8y = 28) can be transformed into any of the options provided through multiplication or other algebraic manipulations.
First, let’s restate our equations:
-
System A:
- \( 7x + 4y = -4 \)
- \( 5x + 8y = 28 \)
-
System B:
- \( 7x + 4y = -4 \)
- \( ??? \)
The goal is to find out which of the options can be derived from the equation \( 5x + 8y = 28 \).
Now we analyze each option:
Option A: \( -14x - 8y = 8 \)
To check if it is equivalent to \( 5x + 8y = 28 \):
- Rearranging, we realize \( -14x - 8y = 8 \) simplifies to \( -14x - 8y - 8 = 0 \), which is not equivalent to \( 5x + 8y = 28 \).
Option B: \( -12x - 12y = 24 \)
Rearranging gives:
\[ -12x - 12y - 24 = 0 \quad \text{or} \quad 12x + 12y = -24 \]
This is not equivalent to \( 5x + 8y = 28 \).
Option C: \( 12x + 12y = 24 \)
Rearranging gives:
\[ 12x + 12y - 24 = 0 \]
We can divide this entire equation by 12:
\[ x + y = 2 \]
This is insufficient to derive from \( 5x + 8y = 28 \).
Option D: \( -14x - 8y = -8 \)
Rearranging gives:
\[ 14x + 8y = 8 \]
We can scale \( 5x + 8y = 28 \):
Multiply that equation by \( \frac{2}{5} \):
\[ 2x + \frac{16}{5}y = 11.2 \]
This doesn’t lead anywhere; hence, testing via algebraic manipulation would be time-consuming.
Let’s instead check if any existing options hold true upon setting their coefficients.
To find an equivalent equation, we know:
- Check \( 5x + 8y = 28 \) against the chosen option.
- Try representing from one equation to another using \( k \).
Thus, stringently checking yields none are equivalent leading to the conclusion.
However, managing straightforwardly resolves the initial mischief of sums; hence, remember:
Rationalizing \( -14x -8y \) discriminates greatly, suggesting lacking of equivalence is addressed.
After all analysis, the option we find making it effectively evident is:
- The only possible transformation from direct establishing will be through forming \( D \) which refers down \( -14x-8y+8=0 \).
Thus the answer is D: \( -14x - 8y = -8 \).