A solution to a system represents values of the contextual quantities that satisfy all stated linear relationships simultaneously; the ordered pair is interpreted with its units and meaning, not merely as an algebraic coordinate. The intersection of two lines may indicate one feasible situation, no feasible situation, or infinitely many equivalent situations, while constraints such as nonnegative amounts, whole-number counts, or realistic bounds may rule out algebraic solutions. The focus is on two-variable linear models, not matrix methods, nonlinear systems, or more abstract solution sets.
A solution to a system tells what the variables mean in the situation, not just where two lines meet. The values must satisfy both equations at the same time and follow any restrictions in the problem, such as nonnegative or whole-number amounts.
Example: A school play sells adult tickets for 5. In all, 120 tickets are sold for a total of $810. How many of each type of ticket were sold?
Define the variables.
Let
Write a system from the information.
The total number of tickets is 120:
The total money collected is $810:
Solve the system.
From the first equation,
Substitute this into the second equation:
Now find :
Interpret the ordered pair.
The algebraic solution is
Since represents adult tickets and represents student tickets, this means:
The school sold 70 adult tickets and 50 student tickets.
Check the meaning in both equations.
Ticket count:
Money collected:
The values are also nonnegative whole numbers, so they make sense in context. The intersection of the two lines represents this one situation that satisfies both requirements.
Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill
Earned ?