Why is the system of equations 2x+7y=14

Question

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

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 1/2 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)
.
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 .

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)
.

Answer 1

To determine why the system of equations

1. \(2x + 7y = 14\) and \(4x + 5y = 10\)

is equivalent to

2. \(4x + 14y = 28\) and \(8x + 10y = 20\),

we should look at how these equations relate to each other.

1. The first equation \(2x + 7y = 14\) can be multiplied by 2 to yield \(4x + 14y = 28\).
2. The second equation \(4x + 5y = 10\) can be multiplied by 2 to yield \(8x + 10y = 20\).

This shows that the first system of equations can be transformed into the second by multiplying each equation by the same factor (2 in this case).

Looking at the provided responses, the correct reason for the equivalence of the two systems is:

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

(Note: The response mentions multiplying by 12, which is not correct, and there are also references to intercepts that do not explain the equivalence of the two systems.)