To determine which equation will make System B equivalent to System A, we first need to analyze System A, which consists of the equations:
- \(7x + 4y = -4\)
- \(5x + 8y = 28\)
From these two equations, we can find out if there is a relationship or a dependency between them.
To check if the second equation can be expressed as a multiple of the first condition in System A, we can solve for one variable in terms of the others. This will help us derive another equation that mirrors the solution set of the first.
Let's manipulate the first equation:
- From \(7x + 4y = -4\), we can express \(y\) in terms of \(x\): \[ 4y = -4 - 7x \quad \Rightarrow \quad y = -1 - \frac{7}{4}x \]
Next, we substitute \(y\) back into the second equation: 2. The second equation is \(5x + 8y = 28\). Substitute \(y\): \[ 5x + 8\left(-1 - \frac{7}{4}x\right) = 28 \] \[ 5x - 8 - 14x = 28 \] \[ -9x - 8 = 28 \quad \Rightarrow \quad -9x = 36 \quad \Rightarrow \quad x = -4 \]
Now that we have one solution \(x = -4\), we can substitute back to find \(y\): \[ y = -1 - \frac{7}{4}(-4) = -1 + 7 = 6 \]
So, one solution of the system is \((-4, 6)\).
Next, we need to check which of the options given can create a situation where both System A and System B yield the same or equivalent solutions.
Option A: \(12x + 12y = 24\)
Simplifying, we divide everything by 12: \[ x + y = 2 \]
Option B: \(-14 - 8y = -8\)
Rearranging gives: \[ -8y = 6 \Rightarrow y = -\frac{3}{4} \]
Option C: \(-14x - 8y = 8\)
This doesn’t present an easy equivalence to the original set without further transforms.
Option D: \(-12x - 12y = 24\)
This simplifies to: \[ x + y = -2 \]
Next, we can analyze which of those transformed equations matches the structure of the original system.
To ensure we derive the same line as previous, we realize that equivalency means it should have a common solution, which checks out with a scaled version (taking multiples).
After checking these options back, only one has a proportional structure tied to \(5x + 8y = 28\):
Let’s put forth the requirement:
From \(12x + 12y = 24\): \[ x + y = 2 \quad \text{(clear)} \]
That suggests, in terms of solutions, we’ll reinforce needing ratios that can mirror:
Proportionally restructuring as guided above, we mean that simply speaking identical forms adhere to those conditions, while still toggling with one another in observation states as multiples.
Thus, the equivalency rests strictly on multiplicands so effectively that A -> B is our relevant output: Therefore, the corresponding answer would be \(A\) or essentially \(12x + 12y = 24\)!
Answer: A. \(12x + 12y = 24\).