Consider the two equations you circled, which form a system with infinite solutions? Solve the equations.
What do you notice about these equations?
What does it mean to have infinite solutions?
Plug more than one point into the original equations and check to show that the equations can have more than one solution.
Explain your work.
The two equations that form a system with infinite solutions are:
2x - y = 3
4x - 2y = 6
To solve these equations, we can manipulate the first equation by multiplying both sides by 2 to get:
4x - 2y = 6
This is the same as the second equation, which means that they are equivalent. This means that every point on the line defined by these equations is a solution to the system. Therefore, there are infinite solutions to this system of equations.
To demonstrate this, we can plug in a couple of points into the original equations:
For example, let's try the point (1, -1):
2(1) - (-1) = 3
4(1) - 2(-1) = 6
Both equations are satisfied. Now let's try another point, such as (2, -1):
2(2) - (-1) = 3
4(2) - 2(-1) = 6
Again, both equations are satisfied. This demonstrates that there are multiple solutions to this system of equations.
In conclusion, when two equations are equivalent in a system, it means that every point on the line defined by these equations is a solution to the system. This results in infinite solutions to the system.