To understand 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 analyze the relationships between the equations.
- Scaling: The first system's equations can be manipulated through multiplication to produce the second system:
- The equation \(2x + 7y = 14\) can be multiplied by 2 to obtain: \[ 4x + 14y = 28 \]
- The equation \(4x + 5y = 10\) can be multiplied by 2 to obtain: \[ 8x + 10y = 20 \]
Thus, we can see that the second system is simply the first system multiplied by 2.
Based on the options provided:
- All equations share the point (0,2): This is incorrect because not all equations correspond to the same intercepts.
- Both equations, 2x + 7y = 14 and 4x + 5y = 10, can be multiplied by 12 to produce the equivalent system of equations: This is incorrect as shown above.
- The intercepts of both 4x + 5y = 10 and 8x + 10y = 20 are (0,2.5) and (2,0): This needs validation, but it doesn't explain the equivalence.
- The intercepts of the equations 2x + 7y = 14 and 4x + 14y = 28 are (0,2) and (7,0): This is misleading because intercepts do not determine equivalence, but they are essentially scaled versions of the original equations.
Given the operations have shown equivalence through multiplication, the most relevant understanding here is that multiplication by a constant factor conservatively relates the two sets of equations.
In conclusion, the best reasoning for why the systems are equivalent is the first option, mainly because you can multiply the first system's equations by 2 to get the second system. However, none of the response explanations directly state that, which may indicate an oversight in the options given.