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 each option.
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Option A: The intercepts of \(2x + 7y = 14\) and \(4x + 14y = 28\) are \((0, 2)\) and \((7, 0)\).
- Setting \(x = 0\) in \(2x + 7y = 14\): \[ 2(0) + 7y = 14 \implies 7y = 14 \implies y = 2 \implies (0, 2) \]
- Setting \(y = 0\) in \(2x + 7y = 14\): \[ 2x + 7(0) = 14 \implies 2x = 14 \implies x = 7 \implies (7, 0) \]
- For \(4x + 14y = 28\), these calculations lead to the same intercepts, but this does not imply equivalence to the other system.
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Option B: Both equations, \(2x + 7y = 14\) and \(4x + 5y = 10\), can be multiplied by \(\frac{1}{2}\) to produce the equivalent system of equations.
- The first equation \(2x + 7y = 14\) multiplied by \(2\) gives \(4x + 14y = 28\).
- The second equation \(4x + 5y = 10\) multiplied by \(2\) gives \(8x + 10y = 20\).
- Therefore, this transformation holds, making this choice valid for demonstrating equivalence.
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Option C: The intercepts of both \(4x + 5y = 10\) and \(8x + 10y = 20\) are \((0, 2.5)\) and \((2,0)\).
- The \(y\)-intercept of \(4x + 5y = 10\) occurs when \(x = 0\): \[ 4(0) + 5y = 10 \implies 5y = 10 \implies y = 2 \implies (0, 2) \]
- The \(x\)-intercept of \(4x + 5y = 10\) occurs when \(y = 0\): \[ 4x + 5(0) = 10 \implies 4x = 10 \implies x = 2.5 \implies (2.5, 0) \]
- Thus, both equations do not share the same intercepts.
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Option D: All equations share the point \((0, 2)\).
- We already found that \((0, 2)\) is a solution for \(2x + 7y = 14\). For the other equations, we can test them:
- \(4x + 5y = 10\) at \((0, 2)\) yields: \[ 4(0) + 5(2) = 10 \quad \text{(True)} \]
- For \(4x + 14y = 28\): \[ 4(0) + 14(2) = 28 \quad \text{(True)} \]
- For \(8x + 10y = 20\): \[ 8(0) + 10(2) = 20 \quad \text{(True)} \]
- All equations do share the point \((0, 2)\).
Because both Option B provides a valid transformation between the two systems, it is a clear and correct reason why the first system is equivalent to the second. Option D is also correct, but it does not provide a transformation method like Option B does.
Therefore, the best answer for why the systems are equivalent is B: "Both equations, \(2x + 7y = 14\) and \(4x + 5y = 10\), can be multiplied by \(1/2\) to produce the equivalent system of equations."