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:
1) 2x - 3y = 6
2) 4x - 6y = 12
To solve these equations, we can see that equation 2 is simply an expansion of equation 1, so they are essentially the same equation. This means that any values of x and y that satisfy the first equation will also satisfy the second equation, and vice versa.
To show that these equations have infinite solutions, we can rewrite the first equation as: 2(x - 3/2y) = 6
Dividing by 2, we get: x - 3/2y = 3
Rearranging, we get: x = 3 + 3/2y
This shows that for any value of y, there is a corresponding x value that will satisfy the equation.
For example, if we take y = 0, then x = 3. This gives the point (3, 0) as a solution to the equations.
Similarly, if we take y = 2, then x = 6. This gives the point (6, 2) as another solution.
In conclusion, having infinite solutions means that there are multiple pairs of values for x and y that satisfy the system of equations. This occurs when the equations are not independent and are essentially linearly dependent, as in this case.