Question

Use the table to answer the question.

System A System B
x+4y=8

2x−5y=29

2x−5y=29

3x−y=37

True or false: The systems in the table are equivalent. Enter 1 for true or 2 for false.

(1 point)
__

Answer 1

the answer is 1 (true)

Answer 2

To determine whether the systems of equations in the table are equivalent, we need to check if they have the same solution(s).

Let's first rewrite the systems for clarity:

**System A:**
1. \( x + 4y = 8 \)
2. \( 2x - 5y = 29 \)

**System B:**
1. \( 2x - 5y = 29 \) (same as the second equation of System A)
2. \( 3x - y = 37 \)

We can see that both systems contain the equation \( 2x - 5y = 29 \), which is the second equation in both systems.

Now we will check whether the first equation in System A (\( x + 4y = 8 \)) has the same solution as the second equation in System B (\( 3x - y = 37 \)).

1. Solve the first equation of System A for one variable and substitute it into the second equation of System B:
\[ x + 4y = 8 \]
→ \( x = 8 - 4y \)

Substituting into the second equation of System B:
\[ 3(8 - 4y) - y = 37 \]
\[ 24 - 12y - y = 37 \]
\[ 24 - 13y = 37 \]
\[ -13y = 37 - 24 \]
\[ -13y = 13 \]
\[ y = -1 \]

Now, substituting \( y = -1 \) back into the equation \( x + 4y = 8 \) to find \( x \):
\[ x + 4(-1) = 8 \]
\[ x - 4 = 8 \]
\[ x = 12 \]

So, one solution from System A is \( (12, -1) \).

2. Now we'll check if \( (12, -1) \) satisfies the equations from System B, particularly:
\[ 3x - y = 37 \]
Substituting \( x = 12 \) and \( y = -1 \):
\[ 3(12) - (-1) = 36 + 1 = 37 \]
This is true.

Since the first equation of System A leads us to the solution \( (12, -1) \) which also satisfies the second equation in System B, we should check the first equation of System B (\( 3x - y = 37 \)) to confirm that it leads to the same conclusion.

As both systems have at least one mutual solution, but the first equations differ, the systems are not equivalent:

Thus, the final answer is **2 (false)**.