Final answer:
Tom's solution to the system of equations involving a parabola and a linear equation is reasonable only if the vertex of the parabola satisfies the linear equation. Upon evaluating, we find that the vertex does not satisfy the linear equation, indicating a need for re-evaluation of the conclusion or changes in the equation. The solution must also be consistent with both equations and contextually make sense.
Step-by-step explanation:
The student is working with a system of equations involving a parabola and a straight line. They're seeking to understand the conditions under which the solutions to the equations are considered reasonable. In this context, for Tom's solution to be reasonable, the vertex of the parabola (equation 1) should coincide with one of the points where the parabola intersects the linear equation (equation 2). In other words, after solving the system of equations, the coordinates of the parabola's vertex should satisfy the linear equation y = -x.
For the problem at hand, equation 1 is already in vertex form, making it easy to identify the vertex as (3, 4). Substituting these values into equation 2, we get 4 = -(3), which is not true. Thus, to satisfy Tom's condition that one solution is at the vertex of the parabola, the linear equation would need to be changed or Tom's conclusion about the situation would need to be re-evaluated. Furthermore, any actual solutions to the system must be real numbers and should be points where the graphs of the two equations intersect.
It's critical to note that the solution methods involve finding the intersection of the two equations by substitution or elimination, checking the solutions for consistency with both original equations, and ensuring the results are within the context of the problem.