1. **State the problem:** We need to find how many solutions exist for the system of linear equations: $$y = 2x + 5$$ $$y = 2x + 4$$ 2. **Recall the rule for solutions of linear systems:** - If the lines have different slopes, they intersect at exactly one point (one solution). - If the lines have the same slope but different intercepts, they are parallel and never intersect (no solution). - If the lines are identical (same slope and intercept), there are infinitely many solutions. 3. **Analyze the given equations:** - Both lines have slope $2$. - The first line has intercept $5$, the second has intercept $4$. 4. **Since slopes are equal but intercepts differ, the lines are parallel and do not intersect.** 5. **Conclusion:** There are **no solutions** to this system because the lines are parallel and never meet.