Model situations with systems of equations or inequalities, represent constraints, and interpret solutions as viable or non-viable (NC.M1.A-CED.3, A-REI.6).

What this topic is asking

NC.M1.A-CED.3 asks you to represent constraints by equations or inequalities and interpret solutions as viable or non-viable, while A-REI.6 supplies the solving. In a modeling problem, two real conditions become two equations (a system), and limits on the variables become inequalities (constraints). You build the system, solve it, and check the answer against reality.

Two conditions become two equations

The signal for a system is that the problem gives two independent facts about two unknowns.

A common pattern: a "how many of each" word problem with both a count total and a money total gives a count equation and a value equation.

Building and solving a model

Constraints and viability

A-CED.3 emphasizes that not every algebraic answer is acceptable. Constraints come in two forms:

• Built into the setup as inequalities: $p \ge 0$, $n \ge 0$, or a budget $2p + 3n \le 30$.
• Checked afterward: a solved value that is negative, or fractional where only whole items make sense, is non-viable.

The test rewards interpreting the solution, not just computing it.

How the NC Math 1 EOC examines this topic

• Gridded response. Set up and solve a system, then enter how many of one item.
• Multiple choice. Choose the system that models a situation, or judge whether a solution is viable.
• Calculator-active. Context problems often sit in the calculator-active section.

Modeling draws together creating equations (one condition at a time), solving systems algebraically, and graphing inequalities for constraint regions.

Why viability is part of the mathematics

It is tempting to think the math ends when you find $x$ and $y$, but modeling has a final, essential step: asking whether the answer can be true. A system does not know that tickets come in whole numbers or that counts cannot be negative; it faithfully reports whatever the equations imply. Interpreting the solution against the context is where a person, not the algebra, must judge. This is why A-CED.3 pairs "represent constraints" with "interpret solutions as viable or non-viable": a model is only useful if its output is read sensibly. On the EOC, a question may deliberately produce a negative or fractional answer to check that you notice it cannot be the real-world solution.

Try this

Q1. Two numbers sum to $20$ and differ by $4$. Write and solve the system. [2 points]

• Cue. $x + y = 20$, $x - y = 4$; add to get $2x = 24$, $x = 12$, $y = 8$.

Q2. A solution gives $4.5$ buses needed. Is it viable? [1 point]

• Cue. Buses are whole; $4.5$ is non-viable, so round up to $5$ in context.