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 Start Fraction 1 over 2 End Fraction to produce the equivalent system of equations.

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

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 .

Answer 1

To determine why the system of equations

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

is equivalent to the system of equations

3. \(4x + 14y = 28\)
4. \(8x + 10y = 20\)

let's analyze the transformations involved.

1. Multiplying equations:
   • The first equation, \(2x + 7y = 14\), can be multiplied by 2 to produce \(4x + 14y = 28\).
   • The second equation, \(4x + 5y = 10\), can be multiplied by 2 to produce \(8x + 10y = 20\).

Since each original equation can be transformed into each new equation by multiplying by the same constant (in this case, 2), the two systems are equivalent.

Therefore, the correct response is:

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