Represent constraints by systems of equations and inequalities and interpret solutions as viable or nonviable options in a modeling context (TN A1.A.CED.A.3).

What this topic is asking

Standard A1.A.CED.A.3 is the modeling end of systems: represent a situation's constraints with a system of equations or inequalities, then solve and decide which solutions are viable. A system of equations pins down an exact answer (how many of each); a system of inequalities describes a feasible region of all acceptable options.

Setting up two equations

The reliable pattern: each independent fact about the situation becomes one equation. A ticket problem has a count fact and a money fact, so it yields two equations in the two unknowns.

Constraints as inequalities

When the situation involves limits rather than exact totals, use inequalities. "No more than 20 hours" is $x + y \le 20$; "at least 150 dollars earned" is an earnings inequality. The feasible region is the overlap, and often a hidden constraint applies: hours and counts are non-negative, so $x \ge 0$ and $y \ge 0$ usually belong to the system even when unstated.

How TNReady examines this topic

• Numeric response. Set up and solve a two-equation system, entering one quantity.
• Multiple choice. Choose the system (equations or inequalities) that models the constraints, with sign-direction distractors.
• Multiple select. Choose all viable combinations from a feasible region.

A clarifying idea is that this standard reuses the solving skills from solving systems and the region skills from graphing systems; the new work is the translation from words and the viability judgement.

Why viability is part of the answer

A system can produce a mathematically correct solution that makes no sense in context, and recognizing that is graded. If a ticket model returned $a = 2.5$ adult tickets, the algebra might be right but the answer is nonviable, you cannot buy half a ticket, so either the problem expects whole numbers (signaling a setup to recheck) or the scenario is impossible as stated. In a feasible-region item, only the integer points inside the region may be viable when the quantities are discrete (people, items, tickets), even though the shaded region contains infinitely many real points. Always close a modeling problem by asking whether the numbers can exist in the real situation, and report the answer with units and meaning, not just a bare value.

Try this

Q1. Two numbers sum to $30$ and differ by $8$. Write a system and find them. [2 points]

• Cue. $x + y = 30$, $x - y = 8$; add to get $2x = 38$, $x = 19$, $y = 11$.

Q2. A vendor needs at least $100$ items total ($x + y$) and at most $400$ dollars spent at $5$ and $3$ dollars each. Write the system. [2 points]

• Cue. $x + y \ge 100$, $5x + 3y \le 400$ (with $x, y \ge 0$).