Question

Answer two questions about Systems
\[A\] and
\[B\]:
System
\[A\]
\[\text{\quad}\] System
\[B\]

\[\begin{cases}-4x-6y=9\\\\3x+y=-4\end{cases}\]

\[\begin{cases}-4x-6y=9\\\\-x-5y=5\end{cases}\]
1) How can we get System
\[B\] from System
\[A\]?
Choose 1 answer:
Choose 1 answer:
(Choice A) Replace only the right-hand side of one equation with the sum/difference of the right-hand sides of both equations
A
Replace only the right-hand side of one equation with the sum/difference of the right-hand sides of both equations
(Choice B) Replace one equation with the sum/difference of both equations
B
Replace one equation with the sum/difference of both equations
(Choice C) Swap only the left-hand sides of both equations
C
Swap only the left-hand sides of both equations
(Choice D) Swap the order of the equations
D
Swap the order of the equations
2) Based on the previous answer, are the systems equivalent? In other words, do they have the same solution?
Choose 1 answer:
Choose 1 answer:
(Choice A) Yes
A
Yes
(Choice B) No
B
No

Answer 1

Let's analyze the systems step by step to answer both questions.

1) To get System \( B \) from System \( A \):

System \( A \) is:
\[
\begin{cases}
-4x - 6y = 9 \\
3x + y = -4
\end{cases}
\]

System \( B \) is:
\[
\begin{cases}
-4x - 6y = 9 \\
-x - 5y = 5
\end{cases}
\]

The first equation in both systems, \(-4x - 6y = 9\), is identical. The second equations differ:
- From System \( A \): \( 3x + y = -4 \)
- From System \( B \): \( -x - 5y = 5 \)

Now, let's see if we can express the second equation of System \( B \) in terms of the two equations from System \( A \):

If we take the sum of the left-hand sides of both equations in System \( A \):
\[
(-4x - 6y) + (3x + y) = 9 + (-4)
\]
This simplifies to:
\[
-4x - 6y + 3x + y = 5 \\
-x - 5y = 5
\]
We can indeed replace the second equation in System \( A \) with the sum of both equations to get the second equation in System \( B\).

Therefore, the correct answer is:
**(Choice B)** Replace one equation with the sum/difference of both equations.

2) Are the two systems equivalent — do they have the same solution?

Since we established that we can transform System \( A \) into System \( B \) using a valid algebraic operation, both systems effectively contain the same information about the variables \( x \) and \( y \). Therefore, they share the same solution.

The answer is:
**(Choice A)** Yes.