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

The intercepts of the equations 2x+7y=14 and 4x+14y=28 are (0,2) and (7,0).
The intercepts of the equations 2 x plus 7 y equals 14 and 4 x plus 14 y equals 28 are left parenthesis 0 comma 2 right parenthesis and left parenthesis 7 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 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 .

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

Answer 1

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 analyze the relationships between the equations.

The first pair of equations can indeed be manipulated.

1. The first equation, \(2x + 7y = 14\), can be multiplied by 2 to yield \(4x + 14y = 28\). This shows that the first equation is equivalent to the first equation in the second system.

2. The second equation, \(4x + 5y = 10\), can be multiplied by 2 to yield \(8x + 10y = 20\). This shows equivalence to the second equation in the second system.

Thus, both original equations can be transformed to match the second system by simple multiplication by constants, indicating the two systems represent the same lines in the coordinate plane.

The correct response to this question is:

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