Asked by Landon Sharp
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? (1 point) Responses −14x−8y=−8 negative 14 x minus 8 equals negative 8 −12x−12y=24 negative 12 x minus 12 y equals 24 −14x−8y=8 negative 14 x minus 8 y equals 8 12x+12y=24 12 x plus 12 y equals 24
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Answered by
GPT-4o mini
To determine which of the given equations will make the two systems equivalent, we need to check if we can convert the equation from System B, \(5x + 8y = 28\), into the form of System A, \(7x + 4y = -4\).
To do this, we can manipulate \(5x + 8y = 28\).
1. We can scale both sides of \(5x + 8y = 28\) to obtain the form that can be compared to \(7x + 4y = -4\).
We want to manipulate the equations to find a form that represents the same line.
However, an easier method to check equivalency is to find if we can get a multiple of the equation of System A from System B.
To compare the equations:
- **Multiply the equation of System A by some constant:**
To see if we can relate it to System B, we notice that \(7x + 4y = -4\) can be manipulated. However, let's focus first on the transformation requirements.
2. **Finding a multiple relationship:**
A common approach is to set up both equations and test the equality condition after scaling or manipulate through elimination or substitution.
3. **Comparing with options:**
Let's quickly analyze each of the response options to understand if they relate:
- **Option 1: \(-14x - 8y = -8\)**
- This can be divided by -2 → \(7x + 4y = 4\) (not equivalent).
- **Option 2: \(-12x - 12y = 24\)**
- Can be reduced but doesn’t relate.
- **Option 3: \(-14x - 8y = 8\)**
- Dividing by -2 leads to \(7x + 4y = -4\) which matches System A.
- **Option 4: \(12x + 12y = 24\)**
- Can be simplified but doesn’t relate.
From the options, the valid choice to ensure equivalency is:
**Response: \(-14x - 8y = 8\)** is the term that makes the systems equivalent as it simplifies directly to match the form of System A.
To do this, we can manipulate \(5x + 8y = 28\).
1. We can scale both sides of \(5x + 8y = 28\) to obtain the form that can be compared to \(7x + 4y = -4\).
We want to manipulate the equations to find a form that represents the same line.
However, an easier method to check equivalency is to find if we can get a multiple of the equation of System A from System B.
To compare the equations:
- **Multiply the equation of System A by some constant:**
To see if we can relate it to System B, we notice that \(7x + 4y = -4\) can be manipulated. However, let's focus first on the transformation requirements.
2. **Finding a multiple relationship:**
A common approach is to set up both equations and test the equality condition after scaling or manipulate through elimination or substitution.
3. **Comparing with options:**
Let's quickly analyze each of the response options to understand if they relate:
- **Option 1: \(-14x - 8y = -8\)**
- This can be divided by -2 → \(7x + 4y = 4\) (not equivalent).
- **Option 2: \(-12x - 12y = 24\)**
- Can be reduced but doesn’t relate.
- **Option 3: \(-14x - 8y = 8\)**
- Dividing by -2 leads to \(7x + 4y = -4\) which matches System A.
- **Option 4: \(12x + 12y = 24\)**
- Can be simplified but doesn’t relate.
From the options, the valid choice to ensure equivalency is:
**Response: \(-14x - 8y = 8\)** is the term that makes the systems equivalent as it simplifies directly to match the form of System A.
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