Slope-Intercept, Point-Slope, and Standard Forms

Objective: Read a slope and intercept from y = mx + b.

Prerequisite skills: slope, coordinate pairs, and linear equation operations. In slope-intercept form, m is the change in y per unit x and b is y when x = 0. A negative intercept is a location, not a negative slope. Common error: confusing the constant term with the slope. Restrictions and interpretation: Vertical lines use x = constant and do not have finite slope.

Objective: Use a point and slope to predict another value.

Point-slope form follows directly from the slope ratio. It anchors the line at a known point and is useful before expanding into other forms. Subtract the known coordinate, including its sign. Common error: using x + x₁ when the point has a positive x-coordinate. Restrictions and interpretation: Vertical lines use x = constant and do not have finite slope.

Objective: Find slope by isolating y in a standard-form equation.

In Ax + By = C with B nonzero, isolate y to get y = −(A/B)x + C/B. The slope includes a minus sign from subtracting Ax. If B = 0 and A ≠ 0, the line is vertical. Common error: reading A as the slope without dividing by B and changing sign. Restrictions and interpretation: Vertical lines use x = constant and do not have finite slope. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.