Georgia Milestones Algebra: a complete guide to linear equations, inequalities, and systems

What this part of the course demands

This guide covers Linear Equations, Inequalities, and Systems, the core of the Patterning and Algebraic Reasoning (A.PAR) domain and, with functions, the largest block of points on the EOC. Fluent linear work is the single most reliable route to the Proficient level. Each dot-point page carries its own worked Milestones-style questions: solving linear equations, solving linear inequalities, graphing linear equations, writing linear equations, solving systems of equations, and linear inequalities in two variables.

Solving linear equations in one variable

Isolate the variable by undoing operations: distribute, clear fractions with the LCD, collect like terms, and divide. The number of solutions depends on the variable terms: if they survive you get one solution; if they cancel and leave a true statement, infinitely many (an identity); if they cancel and leave a false statement, none (a contradiction). Always check by substituting.

Solving linear inequalities in one variable

Solve exactly like an equation, but reverse the inequality sign when you multiply or divide by a negative. Graph the solution on a number line with an open circle for $<$ or $>$ and a closed circle for $\le$ or $\ge$, shading toward the solution. In words, "at most" is $\le$ and "at least" is $\ge$.

Graphing linear equations

From slope-intercept form $y = mx + b$, plot $(0, b)$ and step off the slope. From standard form $Ax + By = C$, find intercepts by setting each variable to zero. From point-slope form $y - y_1 = m(x - x_1)$, plot the point and step off the slope. Two points determine the line; choose ones with integer coordinates for hot-spot accuracy.

Writing linear equations

Use point-slope form as the universal tool. From two points, find $m = \frac{y_2 - y_1}{x_2 - x_1}$ first, then point-slope. From a context, the starting value is the y-intercept and the rate is the slope, and you must interpret both in the units of the problem.

Solving systems of linear equations

A system's solution is the point on both lines. Use substitution when a variable is isolated, elimination when a variable's coefficients match or are easy to match. Parallel lines give no solution; identical lines give infinitely many. Check the point in both equations.

Linear inequalities in two variables

Graph the boundary line (solid for $\le$ or $\ge$, dashed for $<$ or $>$), then shade the half-plane chosen by a test point (the origin when it is off the line). For a system, the solution is the overlap of the shaded regions, and a point qualifies only if it satisfies every inequality.

How this strand is examined

• Numeric entry. Solve an equation, an inequality, or a system, and type the value or ordered pair.
• Hot spot / graphing. Plot a line, select a number-line or half-plane graph, or shade the solution region of a system.
• Multiple choice. Identify slope and intercept, the number of solutions, or whether a point is in a solution set.
• Constructed response. Write and solve a model, or solve a system by a named method and justify the choice.

Check your knowledge

Work these as you would for credit on the EOC.

1. Solve $4(x - 1) = 2x + 6$. (2 points)
2. How many solutions does $3(x + 2) = 3x + 6$ have? (1 point)
3. Solve $-3x + 2 < 11$ and describe the graph. (2 points)
4. Find the intercepts of $5x + 2y = 20$. (1 point)
5. Write the line through $(1, 2)$ and $(4, 11)$ in slope-intercept form. (2 points)
6. Solve $\begin{cases} y = x + 1 \\ 2x + y = 7 \end{cases}$. (2 points)
7. Is $(0, 0)$ a solution of $y < x + 1$? (1 point)