To determine which statement must be true if the point [tex]\((1,4)\)[/tex] lies on the graph of an equation, let's analyze each statement one by one:
Statement A: The values [tex]\(x=1\)[/tex] and [tex]\(y=4\)[/tex] make the equation true.
- Given that the point [tex]\((1, 4)\)[/tex] is on the graph of the equation, substituting [tex]\(x = 1\)[/tex] and [tex]\(y = 4\)[/tex] into the equation should satisfy the equation. This statement is true if the point is on the graph.
Statement B: The values [tex]\(x=1\)[/tex] and [tex]\(y=4\)[/tex] are the only values that make the equation true.
- This statement claims that [tex]\((1,4)\)[/tex] is the unique solution to the equation. It asserts there are no other pairs [tex]\((x, y)\)[/tex] that satisfy the equation. Without specific information about the equation, we cannot confirm the uniqueness of the solution just based on the point [tex]\((1, 4)\)[/tex]. This statement is not necessarily true.
Statement C: There are solutions to the equation for the values [tex]\(x=1\)[/tex] and [tex]\(x=4\)[/tex].
- This statement implies that there are solutions for [tex]\(y\)[/tex] when [tex]\(x = 1\)[/tex] and also when [tex]\(x = 4\)[/tex]. However, knowing just that [tex]\((1, 4)\)[/tex] is a point on the graph, there is no information to guarantee that [tex]\(x = 4\)[/tex] produces a valid solution. Therefore, this statement is not reliable as being necessarily true.
Statement D: The values [tex]\(x=4\)[/tex] and [tex]\(y=1\)[/tex] make the equation true.
- This statement suggests that [tex]\((4, 1)\)[/tex] also satisfies the equation. There's no evidence given that substituting [tex]\(x = 4\)[/tex] and [tex]\(y = 1\)[/tex] into the equation will be valid just based on the point [tex]\((1, 4)\)[/tex]. Therefore, this statement is not necessarily true.
Given the above analysis:
- Statement A is the only statement that must be true: The values [tex]\(x=1\)[/tex] and [tex]\(y=4\)[/tex] make the equation true.
Thus, the correct answer is statement [tex]\( \textbf{A} \)[/tex].
