Question

Which of the following shows a system of equations equivalent to 2x+7y=14 and 4x+5y=10 ?(1 point)
Responses

4x+14y=28 and 8x+10y=10
4 x plus 14 y equals 28 and 8 x plus 10 y equals 10

2x+7y=14 and 8x+10y=20
2 x plus 7 y equals 14 and 8 x plus 10 y equals 20

2x+7y=14 and 2x+5y=5
2 x plus 7 y equals 14 and 2 x plus 5 y equals 5

4x+14y=14 and 4x+5y=10

Answer 1

To determine which of the provided options shows a system of equations equivalent to the original system \(2x + 7y = 14\) and \(4x + 5y = 10\), we can manipulate the original system through multiplication or other equivalent operations.

1. **Original System**:
\[
2x + 7y = 14 \quad (1)
\]
\[
4x + 5y = 10 \quad (2)
\]

Now let's analyze the response options:

1. **Option 1**:
\[
4x + 14y = 28 \quad (3)
\]
\[
8x + 10y = 10 \quad (4)
\]
- (3) is obtained by multiplying (1) by 2.
- (4) cannot be derived from (2) as it implies different coefficients.

2. **Option 2**:
\[
2x + 7y = 14 \quad (5)
\]
\[
8x + 10y = 20 \quad (6)
\]
- (5) is the same as (1).
- (6) can be derived from (2) by multiplying by 2: \(2(4x + 5y) = 2(10)\) which gives \(8x + 10y = 20\).

3. **Option 3**:
\[
2x + 7y = 14 \quad (7)
\]
\[
2x + 5y = 5 \quad (8)
\]
- (7) is the same as (1).
- (8) does not represent (2) or any equivalent form.

4. **Option 4**:
\[
4x + 14y = 14 \quad (9)
\]
\[
4x + 5y = 10 \quad (10)
\]
- (10) is the same as (2).
- (9) can’t be derived from (1) or any other equivalent transformation.

The only option that has both equations equivalent to the original system is **Option 2**:
\[
2x + 7y = 14 \quad \text{and} \quad 8x + 10y = 20.
\]

So, the correct response is:
**2x + 7y = 14 and 8x + 10y = 20.**