10.2 Writing Equations of Lines: Point-Slope, Two Points, Parallel & Perpendicular Lines

Why Writing Equations of Lines Matters on the GED

Beyond graphing an equation you're handed, GED assessment target A.6 ("Connect coordinates, lines, and equations") asks you to go the other direction: given a slope and a point, two points, or a description of a parallel or perpendicular relationship, write the line's equation yourself. These items often appear as fill-in-the-blank or drag-and-drop questions, and they build directly on the slope skills from the previous section.

Point-Slope Form

Point-slope form writes a line's equation using one known point (x₁, y₁) and the slope m:

y − y₁ = m(x − x₁)

This form is useful the moment you know a slope and any single point on the line — you don't need to already know the y-intercept to write a correct equation.

Worked Example

Write the equation, in slope-intercept form, of a line through (2, 5) with slope 3.

• Substitute into point-slope form: y − 5 = 3(x − 2)
• Distribute the 3 across both terms: y − 5 = 3x − 6
• Add 5 to both sides: y = 3x − 1

The most common error here is forgetting to distribute the slope to both terms inside the parentheses — writing y − 5 = 3x − 2 instead of y − 5 = 3x − 6.

Writing an Equation from Two Points

When you're given two points instead of a slope, work in two stages: calculate the slope first, then plug the slope and either point into point-slope form (or directly into slope-intercept form to solve for b).

Worked Example

Write the equation of the line through (-1, 4) and (3, -4).

1. Find the slope: m = (−4 − 4) ÷ (3 − (−1)) = −8 ÷ 4 = −2
2. Plug the slope and one point, say (3, -4), into point-slope form: y − (−4) = −2(x − 3)
3. Simplify: y + 4 = −2x + 6 → y = −2x + 2

Check your work by substituting the other original point into the final equation: at x = -1, y = −2(−1) + 2 = 2 + 2 = 4, which matches the point (-1, 4) exactly. This check catches arithmetic slips before you submit an answer.

Parallel Lines

Parallel lines never intersect because they share the exact same slope but have different y-intercepts. If a line has the equation y = -2x + 7, any line parallel to it must also have a slope of -2 — but a different constant term, since two lines sharing both the same slope and the same y-intercept are actually the same line, not two parallel lines.

Worked Example

Write the equation of a line parallel to y = -2x + 7 that passes through (0, 3).

Since the new line needs slope -2, and its y-intercept is given directly by the point (0, 3) — a point already sitting on the y-axis — the equation is simply y = -2x + 3.

Perpendicular Lines

Perpendicular lines cross at a 90° angle, and their slopes are negative reciprocals of each other — flip the fraction, then change the sign. If one line has slope m, a line perpendicular to it has slope −1/m. Multiplying the two slopes together always gives −1.

Worked Example

What is the slope of a line perpendicular to y = (3/4)x − 2?

Flip 3/4 to get 4/3, then change the sign: the perpendicular slope is -4/3.

Comparing Two Given Equations

The GED sometimes gives two full equations and asks whether the lines are parallel, perpendicular, the same line, or neither. Compare slopes and y-intercepts using this quick reference:

GED Exam Scenario

A city planner needs a road that runs parallel to Main Street, whose equation is y = (1/2)x + 4, but the new road must pass through the intersection at (2, 1). Using point-slope form: y − 1 = (1/2)(x − 2), which simplifies to y = (1/2)x. Recognizing "parallel" as an instruction to match the slope — independent of the specific numbers in the original equation — is the core skill being tested.

Common Traps

• Forgetting to distribute the slope across both terms inside the parentheses of point-slope form.
• Treating two lines with identical equations as "parallel" — they are the same line, not two distinct parallel lines.
• Finding the reciprocal of a slope for a perpendicular line but forgetting to also flip the sign (or vice versa).
• After finding slope from two points, plugging mismatched x- and y-values from different points into point-slope form.