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choose the most logical answer you must choose one of the answers provided.

Which of the following shows a system of equations equivalent to 2x+7y=14 and 4x+5y=10 ?(1 point)
Responses

2x+7y=14 and 8x+10y=20
2 x plus 7 y equals 14 and 8 x plus 10 y equals 20

4x+14y=14 and 4x+5y=10
4 x plus 14 y equals 14 and 4 x plus 5 y equals 10

2x+7y=14 and 2x+5y=5
2 x plus 7 y equals 14 and 2 x plus 5 y equals 5

4x+14y=28 and 8x+10y=10

1 answer

To determine an equivalent system of equations to the original equations \( 2x + 7y = 14 \) and \( 4x + 5y = 10 \), we need to see which pairs of equations can be derived from the original ones through valid algebraic manipulations (such as multiplying or adding equations).

  1. The first option is \( 2x + 7y = 14 \) and \( 8x + 10y = 20 \). The second equation can be obtained by multiplying the second original equation \( 4x + 5y = 10 \) by 2, which maintains equivalence.

  2. The second option is \( 4x + 14y = 14 \) and \( 4x + 5y = 10 \). The first equation does not have the same structure as the given equations; thus, it is not equivalent.

  3. The third option is \( 2x + 7y = 14 \) and \( 2x + 5y = 5 \). The second equation is not equivalent to the original second equation.

  4. The fourth option is \( 4x + 14y = 28 \) and \( 8x + 10y = 10 \). The first equation is not equivalent to the original equations.

The only valid pair that maintains equivalency is the first one.

So the most logical answer is: 2x + 7y = 14 and 8x + 10y = 20.