To determine why the system of equations \( 2x + 7y = 14 \) and \( 4x + 5y = 10 \) is equivalent to the system \( 4x + 14y = 28 \) and \( 8x + 10y = 20 \), we can examine the relationships between the pairs.
- First set of equations:
- \( 2x + 7y = 14 \) can be multiplied by 2 to yield \( 4x + 14y = 28 \).
- \( 4x + 5y = 10 \) can be multiplied by 2 as well, which gives \( 8x + 10y = 20 \).
Thus, the first system can be transformed directly into the second system through multiplication by appropriate constants.
- Analyzing the options:
- The statement about intercepts is irrelevant to equivalence in this context.
- The reference to multiplying by \( \frac{1}{2} \) does not apply since we need to multiply by 2 instead.
- The intercepts mentioned in the third option do not pertain to the equations given and would not define equivalence correctly.
- While the claim that "All equations share the point (0,2)" is true for the equations \( 2x + 7y = 14 \) and \( 4x + 14y = 28 \), this does not encompass the whole system.
Therefore, the best answer is:
Both equations, \( 2x + 7y = 14 \) and \( 4x + 5y = 10 \), can be multiplied by 2 to produce the equivalent system of equations.