The two equations that form a system with infinite solutions are:
3. y = 3x + 1
4. y = 3x + 1
Graph of equations 3 and 4 on a coordinate grid:
(Insert graph with equations 3 and 4 plotted)
What I notice about these graphs is that they lie directly on top of each other. This means that they completely coincide and intersect at every point. For a system of equations to have infinite solutions, it means that the two equations are equivalent and represent the same line. In other words, any point on one line will also be a solution for the other line.
Graphically, the graphs of a system with infinite solutions overlap perfectly, showing that the lines are the same and have an infinite number of common solutions.
Let's plug in some points to show that the system of equations has more than one solution:
For equation 3: y = 3x + 1
- When x = 1, y = 3(1) + 1 = 4
- When x = 2, y = 3(2) + 1 = 7
For equation 4: y = 3x + 1
- When x = 1, y = 3(1) + 1 = 4
- When x = 2, y = 3(2) + 1 = 7
As we can see, for both equations, when x = 1, y = 4, and when x = 2, y = 7. This confirms that there are multiple solutions, and the two equations have infinite solutions because they represent the same line.