Why is the system of equations 2x+7y=14 and 4x+5y=10 equivalent to the system of equations 4x+14y=28 and 8x+10y=20?(1 point)

Question

Responses

All equations share the point (0,2).
All equations share the point left parenthesis 0 comma 2 right parenthesis .

The intercepts of both 4x+5y=10 and 8x+10y=20 are (0,2.5) and (2,0).
The intercepts of both 4 x plus 5 y equals 10 and 8 x plus 10 y equals 20 are left parenthesis 0 comma 2.5 right parenthesis and left parenthesis 2 comma 0 right parenthesis .

Both equations, 2x+7y=14 and 4x+5y=10, can be multiplied by 12 to produce the equivalent system of equations.
Both equations, 2 x plus 7 y equals 14 and 4 x plus 5 y equals 10 , can be multiplied by Start Fraction 1 over 2 End Fraction to produce the equivalent system of equations.

The intercepts of the equations 2x+7y=14 and 4x+14y=28 are (0,2) and (7,0).

Answer 1

To determine why the system of equations \(2x + 7y = 14\) and \(4x + 5y = 10\) is equivalent to the system of equations \(4x + 14y = 28\) and \(8x + 10y = 20\), we need to understand what "equivalent systems of equations" means. Two systems of equations are equivalent if they have the same solutions, which typically occurs through scaling, adding, or subtracting equations.

1. Direct multiplication for equivalence:
   • The equation \(2x + 7y = 14\) can be multiplied by 2 to produce \(4x + 14y = 28\).
   • The equation \(4x + 5y = 10\) can also be multiplied by 2 to produce \(8x + 10y = 20\).

Thus, both initial equations can be scaled to yield the second set of equations.

Given this reasoning, the most accurate response is likely:

Both equations, \(2x + 7y = 14\) and \(4x + 5y = 10\), can be multiplied by 2 to produce the equivalent system of equations.

None of the other options correctly describe the relationship in terms of equivalence.