Let's analyze each statement given that the point \((1, 4)\) is on the graph of the equation. This information indicates that substituting \(x = 1\) and \(y = 4\) into the equation satisfies the equation.
Statement A: There are solutions to the equation for the values \(x = 1\) and \(x = 4\).
- This statement implies two separate solutions: one when \(x = 1\) and another when \(x = 4\). However, we are only given information about the point \((1, 4)\), not \((4, y)\) or \((x, y)\) involving \(x = 4\). Therefore, this statement is not necessarily true based on the given point.
Statement B: The values \(x = 1\) and \(y = 4\) make the equation true.
- Since \((1, 4)\) is on the graph of the equation, by definition, substituting \(x = 1\) and \(y = 4\) satisfies the equation. Therefore, this statement is true.
Statement C: The values \(x = 4\) and \(y = 1\) make the equation true.
- This statement suggests that the point \((4, 1)\) is on the graph, which has not been indicated in the problem. Based on the given information, the point \((4, 1)\) has not been verified. Therefore, this statement is false.
Statement D: The values \(x = 1\) and \(y = 4\) are the only values that make the equation true.
- This statement implies that no other point can satisfy the equation except \((1, 4)\). However, with only one point given, we cannot determine whether it is the only solution without more information about the equation. Therefore, this statement cannot be confirmed with the given information.
Based on this analysis, the statement that must be true is:
B. The values [tex]\(x = 1\)[/tex] and [tex]\(y = 4\)[/tex] make the equation true.
