NC Math 1: a complete guide to systems of equations and inequalities

What this strand demands

This guide covers systems of equations and inequalities on the NC Math 1 EOC, drawing on Reasoning with Equations and Inequalities (NC.M1.A-REI.5, A-REI.6, A-REI.11) and the modeling standard A-CED.3. These sit in the Algebra block. The strand is about handling two conditions at once: two equations meeting at a point, or constraints carving out a region. Each dot-point page has its own practice: solving systems algebraically, solving systems by graphing, why equivalent systems work, graphing linear inequalities, and modeling with systems.

Solving systems algebraically

A system of two linear equations is solved by the ordered pair that satisfies both. Substitution: solve one equation for a variable, substitute into the other, solve, and back-substitute. Elimination: add or subtract the equations (scaling first if needed) so one variable cancels, solve, and back-substitute. If the variables cancel with a false statement, there is no solution; with a true statement, infinitely many.

Solving by graphing

The solution is the intersection point of the two lines, because it lies on both. A-REI.11 explains why the x-coordinate of the intersection of $y = f(x)$ and $y = g(x)$ solves $f(x) = g(x)$: at that point the two outputs are equal. Different slopes give one solution; parallel lines give none; identical lines give infinitely many.

Why equivalent systems work

A-REI.5 justifies elimination: replacing one equation with the sum of it and a multiple of the other keeps the same solution set. The reason is that a point satisfying both equations satisfies any combination of them, and the step is reversible, so no solutions are gained or lost. The multiplier must be nonzero.

Graphing linear inequalities

A two-variable inequality graphs as a half-plane. Draw the boundary line solid for $\le$ or $\ge$ and dashed for $<$ or $>$, then use a test point (usually the origin) to decide which side to shade.

Modeling with systems and constraints

Two real conditions become two equations; limits become inequality constraints. Solve, then judge viability: reject negative counts or impossible fractions. A "how many of each" problem with a count total and a money total is the classic system.

How this strand is examined

• Gridded response. Solve a system and enter a value. Exact-match scoring.
• Multiple choice and multiple select. Choose a solution, a modeling system, or whether a solution is viable.
• Technology-enhanced. Plot lines and identify an intersection, or graph an inequality's half-plane.

Check your knowledge

Work these as you would for credit on the EOC.

1. Solve $y = x - 2$ and $2x + y = 10$. (2 points)
2. Solve $3x + y = 9$ and $3x - y = 3$. (2 points)
3. Where do $y = 2x + 1$ and $y = -x + 7$ intersect? (2 points)
4. Two distinct lines have the same slope. How many solutions? (1 point)
5. Is the boundary of $y \ge x - 4$ solid or dashed? (1 point)
6. For $x + y \le 6$, is $(0, 0)$ a solution? (1 point)
7. Adult tickets cost \10$, child$\$6$. A group buys $5$ tickets for \38$. How many of each? (2 points)
8. A solved system gives $-3$ chairs. What do you conclude? (1 point)