Question

Use the table to answer question. System A: 7x + 4y = -4, 5x + 8y = 28 System B: 7x + 4y = -4, ?. Which of the following shows the equation that creates a true equivalent system?

1. -12x - 12y = 24
2. 12x + 12y = 24
3. -14x - 8y = 8
4. -14x - 8y = -8

Answer 1

To find a true equivalent system for System B that includes the equation \(7x + 4y = -4\) from System A, we can look for a new equation that is a scalar multiple of a linear combination of the original equations of System A.

Let's analyze the options provided:

1. **Option 1:** \(-12x - 12y = 24\)

We can simplify this equation by dividing everything by -12:
\[
x + y = -2
\]
This does not appear to relate to any known equivalent action on \(7x + 4y = -4\).

2. **Option 2:** \(12x + 12y = 24\)

Similar to option 1, we can simplify this by dividing everything by 12:
\[
x + y = 2
\]
This equation also does not correlate with any scalar multiple of \(7x + 4y = -4\).

3. **Option 3:** \(-14x - 8y = 8\)

If we divide this whole equation by -2, we get:
\[
7x + 4y = -4
\]
which is indeed the same as one of the equations in System A.

4. **Option 4:** \(-14x - 8y = -8\)

Simplifying this equation by dividing everything by -2 would result in:
\[
7x + 4y = 4
\]
which is different from our original equation from System A.

From this analysis, **Option 3** (\(-14x - 8y = 8\)) yields an equivalent equation to \(7x + 4y = -4\).

Thus, the equation that creates a true equivalent system is:
**Option 3: \(-14x - 8y = 8\)**.