Ohio Algebra I: a complete guide to systems of equations and inequalities

What this category demands

This guide covers systems of equations and inequalities, the Ohio Algebra I block drawn from A-REI (reasoning with equations and inequalities) and A-CED (creating equations). A system asks for the point, or the region, that satisfies two or more conditions at once. It sits in the Expressions and Equations category and can appear with or without a calculator. Each dot-point page has its own practice: solving systems algebraically, solving systems by graphing, graphing linear inequalities, systems of linear inequalities, and modeling with systems.

Solving systems exactly

To solve a linear system, use substitution when a variable is already isolated, or elimination when coefficients line up to cancel. Both find the ordered pair where the lines meet.

Solving by graphing

Graph each line and read the intersection as the solution. Different slopes cross once; same slope, different intercept are parallel (no solution); identical lines overlap (infinitely many). Graphing is exact when the crossing is a clean lattice point and estimates otherwise.

Graphing a single inequality

A two-variable inequality shades a half-plane. Draw the boundary solid for $\leq$ or $\geq$ and dashed for $<$ or $>$, then shade the side a test point (usually the origin) makes true.

Systems of inequalities and modeling

The solution of a system of inequalities is the overlap of the half-planes, the points satisfying every inequality. Adding $x \geq 0$ and $y \geq 0$ can close the overlap into a feasible region. To model a context, define two variables, write one relationship per fact (count and value, or one constraint per limit), solve, and interpret with units.

How this category is examined

• Equation and numeric entry. Solve a system for an ordered pair or a modeled quantity.
• Graphing. Plot lines and click the intersection, or shade a half-plane or a feasible region.
• Multiple choice and multiple-select. Count solutions, match a system to a graph, or pick feasible points.

Check your knowledge

Work these as you would for credit on the Ohio test.

1. Solve $y = 4x$ and $2x + y = 18$. (2 points)
2. Solve $3x + 2y = 1$ and $-3x + y = 5$ by elimination. (2 points)
3. How many solutions does $x + y = 4$ and $2x + 2y = 9$ have? (1 point)
4. For $y \geq -x + 2$, is the boundary solid or dashed, and does $(0, 0)$ lie in the solution? (2 points)
5. Is $(1, 1)$ a solution of the system $y < 2x$ and $y \geq 0$? (2 points)
6. Two numbers differ by $4$ and add to $20$. Find them. (2 points)
7. A plan needs $x \geq 0$, $y \geq 0$, $x + 2y \leq 12$. Is $(4, 3)$ feasible? (1 point)