Why is the system of equations 2x+7y=14 and 4x+5y=10 equivalent to the system of equations 4x+14y=28 and 8x+10y=20 ?(1 point)

Question

Responses

1.Both equations, 2x+7y=14 and 4x+5y=10, can be multiplied by 1/2 to produce the equivalent system of equations.
2.The intercepts of both 4x+5y=10 and 8x+10y=20 are (0,2.5) and (2,0).
3.The intercepts of the equations 2x+7y=14 and 4x+14y=28 is (0,2) and (7,0).
4. All equations share the point (0,2)

**Number 3 is correct, but I do not know if that makes them equivalent**

Answer 1

To determine why the system of equations \(2x + 7y = 14\) and \(4x + 5y = 10\) is equivalent to the system \(4x + 14y = 28\) and \(8x + 10y = 20\), let's analyze their relationships.

1. First System:

   • First equation: \(2x + 7y = 14\)
   • Second equation: \(4x + 5y = 10\)

2. Second System:

   • First equation: \(4x + 14y = 28\)
   • Second equation: \(8x + 10y = 20\)

Now, let's manipulate the first system to see if we can derive the second system using equivalent transformations:

• For the first equation \(2x + 7y = 14\), if we multiply both sides by 2, we get: \[ 4x + 14y = 28 \]
• For the second equation \(4x + 5y = 10\), if we multiply both sides by 2 as well, we obtain: \[ 8x + 10y = 20 \]

Thus, both equations in the first system can be transformed to produce the equations in the second system. This confirms that the two systems are equivalent.

So, the correct response to why these systems are equivalent is:

1. Both equations, \(2x + 7y = 14\) and \(4x + 5y = 10\), can be multiplied by 2 to produce the equivalent system of equations.

While option 3 mentions intercepts of the equations, it does not correctly establish their equivalence. Instead, it's the fact that we can transform the first system into the second by multiplication that shows they are equivalent.

Answer 2

So by multiplying it by 1/2 or one over 2, is a way to make them equivalent?

Answer 3

Multiplying the equations by \(1/2\) (or \(\frac{1}{2}\)) would not make the original system equivalent to the second system you provided. Instead, multiplying by \(2\) (not \(1/2\)) establishes the connection.

Here’s a clearer breakdown:

1. Original System:

   • \(2x + 7y = 14\)
   • \(4x + 5y = 10\)

2. Transformation to the Second System:

   • When we multiply the first equation \(2x + 7y = 14\) by \(2\), we get: \[ 4x + 14y = 28 \]
   • When we multiply the second equation \(4x + 5y = 10\) by \(2\), we get: \[ 8x + 10y = 20 \]

Thus, multiplying by \(2\) gives us the second system:

• \(4x + 14y = 28\)
• \(8x + 10y = 20\)

On the other hand, if you multiply the original system by \( \frac{1}{2} \), you'll end up with a different system:

• \(x + \frac{7}{2}y = 7\)
• \(2x + \frac{5}{2}y = 5\)

This isn’t equivalent to the second system.

So the key point is that multiplying by \(2\) yields the equivalent system, while multiplying by \(1/2\) does not.

Answer 4

Thank you for your answer, but you did not provide an answer. For number one, they stated "when multiplied by 1 / 2 " not 2.

Why is the system of equations 2x+7y=14 and 4x+5y=10 equivalent to the system of equations 4x+14y=28 and 8x+10y=20 ?(1 point)
Responses

1.Both equations, 2x+7y=14 and 4x+5y=10, can be multiplied by 1/2 to produce the equivalent system of equations.
2.The intercepts of both 4x+5y=10 and 8x+10y=20 are (0,2.5) and (2,0).
3.The intercepts of the equations 2x+7y=14 and 4x+14y=28 is (0,2) and (7,0).
4. All equations share the point (0,2)

Answer 5

I understand the confusion, and thank you for your patience. Let’s examine the statements provided in the context of how we can determine equivalence between the two systems of equations.

Original System:

1. \(2x + 7y = 14\)
2. \(4x + 5y = 10\)

Transformed System:

1. \(4x + 14y = 28\)
2. \(8x + 10y = 20\)

To clarify:

1. The transformation from the original system to the second system cannot be achieved by multiplying by \( \frac{1}{2} \). Instead, you could multiply each equation in the original system by \(2\) to reach the second system.

Now let’s analyze the provided response options:

1. Option 1: "Both equations, \(2x+7y=14\) and \(4x+5y=10\), can be multiplied by \(1/2\) to produce the equivalent system of equations."

   • This statement is incorrect because multiplying by \( \frac{1}{2} \) does not yield the equations in the second system.

2. Option 2: "The intercepts of both \(4x + 5y = 10\) and \(8x + 10y = 20\) are \((0, 2.5)\) and \((2, 0)\)."

   • This statement doesn't help establish equivalence between the two original systems.

3. Option 3: "The intercepts of the equations \(2x + 7y = 14\) and \(4x + 14y = 28\) is \((0, 2)\) and \((7, 0)\)."

   • While it mentions intercepts, it doesn't address the equivalence directly and is not relevant to explaining why the systems are equivalent.

4. Option 4: "All equations share the point \((0, 2)\)."

   • This could indicate that there is a common point, but it doesn't demonstrate that the entire systems are equivalent.

Conclusion:

None of the provided options accurately explain why the two systems are equivalent.

To summarize, the correct reason (not given in the options) is that:

• You can derive \(4x + 14y = 28\) and \(8x + 10y = 20\) by multiplying the original equations by \(2\).

If I had to select the option that is the most relevant to their equivalence based on the context given, you could argue that option 4 (that all equations share the point \((0, 2)\)) suggests a common solution, but it does not fully capture the concept of equivalency for the entire systems.

I apologize for the earlier confusion, and thank you for your understanding!