Question

Which system of equations has infinitely many solutions?
4 x + 2 y = 5. Negative 4 x minus 2 y = 1.
Negative 10 x + y = 4. 10 x minus y = negative 4.
Negative 8 x + y = 2. 8 x minus y = 0.
Negative x + 2 y = 6. 7 x minus 2 y = 12.

Answer 1

To determine which system of equations has infinitely many solutions, we need to identify a pair of equations that are equivalent - that is, one can be obtained from the other by multiplying or dividing by a constant.

Let's analyze each option:

1. **4x + 2y = 5 and -4x - 2y = 1**
- These two equations are not equivalent, as rearranging one does not yield the other.

2. **-10x + y = 4 and 10x - y = -4**
- If we multiply the first equation by -1, we get:
\[10x - y = -4\]
This is exactly the second equation in the pair, indicating that these two equations are equivalent. This means that this system has infinitely many solutions.

3. **-8x + y = 2 and 8x - y = 0**
- Rearranging the second gives \(y = 8x\), which is not equivalent to the first equation. Thus, they do not represent the same line.

4. **-x + 2y = 6 and 7x - 2y = 12**
- These equations are not equivalent and they do not represent the same line.

Based on this evaluation, the system of equations that has infinitely many solutions is:

**-10x + y = 4 and 10x - y = -4.**