Final answer:
The equation 0 = 0 after using the linear combination method on a system of equations indicates that there are infinitely many solutions, as both equations represent the same line. The correct answer is option a.There are 0 solutions to the system.
Step-by-step explanation:
When you arrive at an equation like 0 = 0 after applying the linear combination method to a system of equations, it indicates something specific about the solutions to the system. This conclusion originates from manipulating the given equations to find a solution for the variables x and y.
In the case of the system:
4x + 10y = 12
10x + 25y = 30
If through manipulation (like adding, subtracting, or multiplying equations to eliminate variables) you get 0 = 0, this means that the two equations are not independent but rather the same line represented differently. Therefore, the correct answer is d. There are infinitely many solutions to the system. This is because every point on the line described by the first equation is also on the line described by the second equation - the lines coincide.
The other options can be ruled out a priori because we know that:
- a. There are 0 solutions to the system would mean the equations represent parallel lines, which would lead to a result such as 0 = a nonzero number.
- b. The solution to the system is (0, 0) would be the case only if both original equations intersected at the origin.
- c. There are solutions to the system at the x- and y-intercepts does not necessarily follow from the 0 = 0 result as this deals with specific points rather than the totality of all points that satisfy the equations.
To summarize, 0 = 0 in a system of equations during linear combination suggests infinite solutions since the two lines are essentially the same, resulting in their intersection at every point on the line.