NC Math 1: a complete guide to linear equations and functions

What this strand demands

This guide covers linear equations and functions on the NC Math 1 EOC, drawing on Reasoning with Equations and Inequalities (NC.M1.A-REI), Creating Equations (NC.M1.A-CED), and Constructing Linear Functions (NC.M1.F-LE, F-BF). These standards sit in the Algebra and Functions block, the largest part of the test. The linear thread, solving, modeling, rearranging, and graphing, recurs throughout the course and is the most reliable place to bank points. Each dot-point page has its own practice: solving linear equations, solving linear inequalities, creating equations and inequalities, rearranging literal equations, slope and writing linear equations, and graphing linear equations.

Solving linear equations and inequalities

To solve a linear equation, isolate the variable with the properties of equality: distribute, clear fractions, gather variables on one side and constants on the other, then divide by the coefficient. A-REI.1 asks you to justify each step. Inequalities follow the same routine with one extra rule: multiplying or dividing by a negative flips the sign. Graph the result with an open circle ($<$, $>$) or closed circle ($\le$, $\ge$) and a shaded ray. Watch the special cases: variable cancels with a true statement (infinitely many) or a false statement (no solution).

Creating equations from context

The Creating Equations standards reverse solving: build the model from words. Define the variable first, then translate, rate to coefficient, fixed amount to constant, "at least" to $\ge$, "no more than" to $\le$. Two-variable equations describe relationships; inequalities describe constraints. After solving, judge viability: reject negative or fractional answers that cannot exist in context.

Rearranging formulas

A literal equation is solved for a chosen variable by treating the other letters as constants. The algebra is identical to solving with numbers: undo operations in reverse order, and divide the whole opposite side, not one term. If the target appears in two terms, factor it out first. One formula serves many questions: $d = rt$ becomes $t = \frac{d}{r}$ or $r = \frac{d}{t}$.

Slope and writing lines

Slope is the constant rate of change, $m = \frac{y_2 - y_1}{x_2 - x_1}$. To write a line you need a slope and a point: use slope-intercept form $y = mx + b$ with the y-intercept, or point-slope form $y - y_1 = m(x - x_1)$ with any point. From two points, find the slope first, then substitute one point.

Graphing lines

From slope-intercept form, plot $(0, b)$ and step off the slope. From standard form $Ax + By = C$, find the two intercepts (set $y = 0$ for the x-intercept, $x = 0$ for the y-intercept). A horizontal line $y = c$ has slope $0$; a vertical line $x = c$ has undefined slope and is not a function.

How this strand is examined

• Gridded response. Solve an equation, rearrange a formula and compute, or enter a slope or intercept. Exact-match scoring.
• Multiple choice and multiple select. Solve, count solutions, choose a modeling inequality, or match a line to its equation.
• Technology-enhanced. Plot a line or place a number-line solution.

Check your knowledge

Work these as you would for credit on the EOC.

1. Solve $5(x - 2) = 2x + 8$. (2 points)
2. How many solutions does $4x + 1 = 4x - 3$ have? (1 point)
3. Solve $-3x + 5 > 14$. (2 points)
4. A printer costs \80$plus$\$0.04$ per page. Write a cost equation for $p$ pages. (1 point)
5. Solve $A = lw$ for $w$. (1 point)
6. Find the slope through $(2, 1)$ and $(5, 10)$. (1 point)
7. Write the line with slope $-2$ through $(0, 5)$. (1 point)
8. Find both intercepts of $2x + 3y = 6$. (2 points)