Question

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=28 and 8x+10y=10
4 x plus 14 y equals 28 and 8 x plus 10 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=14 and 4x+5y=10
4 x plus 14 y equals 14 and 4 x plus 5 y equals 10

Answer 1

To find a system of equations equivalent to the given equations \(2x + 7y = 14\) and \(4x + 5y = 10\), we can look for equivalent expressions that involve simple manipulations like multiplication or addition of equations.

1. The first option:
\[
2x + 7y = 14 \quad \text{and} \quad 8x + 10y = 20
\]
- The second equation can be derived by multiplying \(4x + 5y = 10\) by 2:
\[
2(4x + 5y = 10) \Rightarrow 8x + 10y = 20
\]
- So this is valid.

2. The second option:
\[
4x + 14y = 28 \quad \text{and} \quad 8x + 10y = 10
\]
- The first equation can be derived by multiplying \(2x + 7y = 14\) by 2:
\[
2(2x + 7y = 14) \Rightarrow 4x + 14y = 28
\]
- However, the second equation \(8x + 10y = 10\) does not correspond to any valid operation from the original equations because it cannot be derived from \(4x + 5y = 10\).

3. The third option:
\[
2x + 7y = 14 \quad \text{and} \quad 2x + 5y = 5
\]
- The second equation does not correspond to the equations given.

4. The fourth option:
\[
4x + 14y = 14 \quad \text{and} \quad 4x + 5y = 10
\]
- The first equation \(4x + 14y = 14\) cannot be derived from the original equations.

Thus, the only correct set of equations that is equivalent to the original system is:

**\(2x + 7y = 14\) and \(8x + 10y = 20\)**.