Let's address each scenario one by one.
**No solution:**
For a system of linear equations to have no solution, the lines represented by the equations must be parallel but not coincident. This means they have the same slope but different y-intercepts. The equation you have is y - 8x = 1, which is the same as y = 8x + 1. Here, the slope is 8, and the y-intercept is 1.
To get a parallel line with no solution, we need an equation with the same slope but a different y-intercept. A new equation with a slope of 8 that does not intersect y = 8x + 1 could be, for instance, y = 8x + 3. Written in the standard form it would look like y - 8x = 3.
**One solution:**
For a system of equations to have exactly one solution, the two lines must intersect at exactly one point. This means that their slopes must be different (they are not parallel). The new equation must not have the slope of 8. A simple way to create such an equation would be to choose any slope that is not 8 and any y-intercept. For example, an equation with a slope of -8 could work, such as y = -8x + 2. In standard form, that's y + 8x = 2.
**Infinitely many solutions:**
For a system to have infinitely many solutions, the two equations must represent the same line; they are coincident. This means the second equation must be a multiple of the first equation. We can take the original equation y - 8x = 1 and simply multiply both sides by the same non-zero number to get an equivalent equation.
For example, if we multiply both sides by 2, we get 2y - 16x = 2, which simplifies to y - 8x = 1 when divided by 2. So this new equation represents the same line, confirming infinitely many solutions.
In summary, based on the condition:
- **No solution:** A possible second equation is y - 8x = 3.
- **One solution:** A possible second equation is y + 8x = 2.
- **Infinitely many solutions:** A possible second equation is 2y - 16x = 2.
