A solution to a system in context is an ordered pair whose coordinates represent two related quantities, with units and meaning determined by the situation, and that satisfies both conditions simultaneously. The learner interprets a single solution as one feasible combination, no solution as incompatible conditions, and infinitely many solutions as conditions that describe the same relationship, while recognizing that practical restrictions such as nonnegative or whole-number quantities may limit which algebraic solutions are meaningful. Advanced modeling cases and more general systems are not included.
A solution to a system in context is an ordered pair that tells what both quantities are. The pair must satisfy both equations and make sense in the situation.
At a school concert, adult tickets cost $9 and student tickets cost $6. A total of 8 tickets are sold for $54. How many of each type of ticket were sold?
Let
The ordered pair will be , so its first coordinate represents adult tickets and its second coordinate represents student tickets.
The total number of tickets is 8:
The total cost is $54:
So the system is
From the first equation,
Substitute this into the second equation:
Simplify:
Now find :
The solution is
Because represents adult tickets and represents student tickets, means:
2 adult tickets and 6 student tickets were sold.
Check both conditions:
The solution is meaningful because the quantities are nonnegative whole numbers.
If a system has no solution, the conditions are incompatible, so no combination works. If it has infinitely many solutions, the two conditions describe the same relationship, so many combinations work. In a real situation, restrictions such as nonnegative or whole-number quantities still determine which solutions make sense.
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