When solving a system of equations by substitution and substituting an expression for one of the variables results in an equation such as [tex]\(7 = 9\)[/tex], this provides a critical insight into the nature of the system of equations.
Here's a step-by-step breakdown of what happened and what Vineet can conclude:
1. Understanding the Contradiction:
- By substituting an expression for one variable, Vineet worked through the equations and ended up with a statement, [tex]\(7 = 9\)[/tex], which is clearly false.
- This false statement indicates that there is no possible set of values for the variables that can satisfy both equations simultaneously.
2. Interpretation of the Result:
- If solving a system of equations leads to a contradiction like [tex]\(7 = 9\)[/tex], it means that the system is inconsistent.
- An inconsistent system of equations does not have any solutions because the equations represent lines (or planes, etc., in higher dimensions) that do not intersect at any point.
3. Conclusion:
- Given the incorrect statement derived from the substitution, Vineet can conclude that there are no values for the variables that satisfy both equations at the same time.
Therefore, Vineet should conclude that the system of equations does not have a solution.
In summary:
- The contradiction [tex]\(7 = 9\)[/tex] indicates inconsistency in the system of equations.
- An inconsistent system has no solution.
Thus, among the given options, the correct conclusion is:
The system of equations does not have a solution.
