Question

Use the table to answer the question.

System A System B
7x+4y=−4

5x+8y=28

7x+4y=−4

?

Which of the following terms will make the systems equivalent?

Answer 1

To determine what term must be added to System B for the two systems to be equivalent, we need to compare the equations from System A and System B and ensure they describe the same line.

From System A, we have:

1. \( 7x + 4y = -4 \)

From System B, we have the first equation:

2. \( 5x + 8y = 28 \)

We need to check if these two equations can produce the same line.

Let’s rearrange equation 2 from System B into a form that allows comparison with equation 1 from System A. We can try to solve \(5x + 8y = 28\) for \(y\):

\[
8y = 28 - 5x
\]
\[
y = \frac{28 - 5x}{8}
\]

Now simplifying this equation gives us a slope-intercept form, but for comparison with System A, we would ultimately want to express it in a similar standard form.

Next, let's analyze the first equation from System A, rewritten in terms of \(y\):

\[
4y = -4 - 7x
\]
\[
y = -\frac{7}{4}x - 1
\]

Now we have both in the form of \(y = mx + b\):
- For System A: the slope is \(-\frac{7}{4}\) and the y-intercept is \(-1\).
- For System B: we can express \(y\) in terms of \(x\) as we calculated.

We need to find a second equation for System B, which must also match the same line as the first equation in System A. To keep things clear, we could multiply the entire first equation from System A by a factor so that we can build an equivalent equation for System B.

Alternatively, to find a term that will make System B equivalent to System A, we can look for relationships between the two equations. If we multiply the equation from System A by some factor:

Let’s multiply the equation \( 7x + 4y = -4 \) by \( k \):

\[
k(7x + 4y) = k(-4)
\]

This is equivalent to matching \(5x + 8y = 28\) via another multiple. So, if we let \(k = 1\) for the first equation and find that \(5x + 8y = 28\) does not produce an equivalent second equation, we need to investigate a way to set the right terms.

Thus, we can say the term which needs to be added to System B could indeed hinge on establishing a common slope or intercept, revealing an equation can be generated following proportional relationships and equivalent restructuring of systems.

One potential term could be examining the inconsistency points from trials:

If by trial or graphing, we determine they intersect or pass equivalently, the system will expose its relational structure yielding blank conditions.

Given a choice between terms, if "8y" is presented, consider it an attempt towards aligning the second equation. Thereby, if it's absent, postulating based on intercepts could stabilize \(y\) at various values. The derived adjustments open vectors towards intersections.

Hence to conclude, without provided specific choices for a term from the systems, a placeholder or variable adjustment satisfying structural equivalency must be identified by testing outputs along integrated methods of equal relations across coefficients and constants.

If you have specific terms to analyze further, I'd be happy to guide you with those!