To determine which option shows a system of equations equivalent to the given equations \(2x + 7y = 14\) and \(4x + 5y = 10\), we can manipulate each equation or check each provided option for equivalence.
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Starting equations: \[ 2x + 7y = 14 \quad (1) \] \[ 4x + 5y = 10 \quad (2) \]
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Multiply equation (1) by a factor (for example, by 2): \[ 2(2x + 7y) = 2(14) \implies 4x + 14y = 28 \quad (3) \]
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Multiply equation (2) by a factor (for example, by 2): \[ 2(4x + 5y) = 2(10) \implies 8x + 10y = 20 \quad (4) \]
Now, let's analyze each answer option to see if they match the forms we've acquired or are equivalent in some way:
A. \(4x + 14y = 28\) and \(8x + 10y = 20\)
- \((3)\) and \((4)\) are correct.
B. \(4x + 14y = 14\) and \(4x + 5y = 10\)
- The first equation is not equivalent since it should be \(4x + 14y = 28\).
C. \(2x + 7y = 14\) and \(2x + 5y = 5\)
- The first equation is correct, but the second is not equivalent to \(4x + 5y = 10\).
D. \(2x + 7y = 14\) and \(8x + 10y = 20\)
- The first is correct, and the second is equivalent since it comes from multiplying \(4x + 5y = 10\) by 2.
Thus, the equivalent systems of equations that match our transformations are:
A. \(4x + 14y = 28\) and \(8x + 10y = 20\)
D. \(2x + 7y = 14\) and \(8x + 10y = 20\)
Since only one option can be chosen and \(D\) shows one of the original equations, the best answer is:
D. \(2x + 7y = 14\) and \(8x + 10y = 20\)