Parallel and Perpendicular Linear Functions

Two lines are parallel if they have the same slope, and perpendicular if the product of their slopes is −1. In brief: y = 2x + 1 and y = 2x − 5 are parallel, while y = 2x + 1 and y = −(1/2)x + 3 are perpendicular, because 2 · (−1/2) = −1. Vertical and horizontal lines are a separate special case.

These two rules let you write the equation of a line when you know a point on it and its relation to another line. They are used for fences, roads, tangents and normals. This guide explains why the rules hold, how to apply them step by step and how to check your answers with a table of slope pairs.

When are two lines parallel?

Two distinct lines y = m₁x + b₁ and y = m₂x + b₂ are parallel exactly when m₁ = m₂ and b₁ ≠ b₂. Equal slopes mean equal steepness, so the lines never meet. If the slopes and the intercepts are both equal, the lines coincide and are the same line.

Example: 3x − 2y = 6 can be written as y = (3/2)x − 3, so its slope is 3/2. The line y = 1.5x + 4 has the same slope and a different intercept, so the two lines are parallel.

When are two lines perpendicular?

Two lines with slopes m₁ and m₂ (both nonzero) are perpendicular exactly when m₁ · m₂ = −1. The slopes are negative reciprocals: m₂ = −1/m₁.

Why it works: rotating a line by 90° swaps its run and rise and changes one sign. A step of (1, m) becomes (−m, 1), whose slope is 1/(−m) = −1/m. For example, the direction (1, 3) rotated becomes (−3, 1), with slope −1/3.

Examples: the negative reciprocal of 4 is −1/4; of −2/3 it is 3/2; of 1 it is −1.

How do you find a parallel or perpendicular line through a point?

1. Find the slope of the given line.
2. Use the same slope (parallel) or its negative reciprocal (perpendicular).
3. Write the equation using point-slope form with the given point.
4. Simplify to y = mx + b.

Example 1: parallel

Find the line parallel to y = −3x + 2 through (2, 5). The slope is −3. Then y − 5 = −3(x − 2), so y = −3x + 11. Check: at x = 2, y = −6 + 11 = 5 ✓.

Example 2: perpendicular

Find the line perpendicular to 2x + 4y = 8 through (−1, 3). Solve for y: y = −(1/2)x + 2, with slope −1/2. The perpendicular slope is 2. Then y − 3 = 2(x + 1), so y = 2x + 5. Check the product: 2 · (−1/2) = −1 ✓.

Example 3: testing three lines

Lines: L1: y = 2x + 1, L2: 4x − 2y = 7 and L3: x + 2y = 4. The slopes are 2, 2 (because y = 2x − 3.5) and −1/2. L1 and L2 are parallel, and L3 is perpendicular to both.

Slopes of parallel and perpendicular lines

What about horizontal and vertical lines?

A horizontal line y = c has slope 0 and a vertical line x = d has no slope. They are perpendicular to each other, but the rule m₁m₂ = −1 does not apply, since 0 has no reciprocal. Two vertical lines are parallel, as are two horizontal lines. The line through (4, 2) perpendicular to y = 7 is x = 4.

Where do these rules come from?

Euclid defined parallel lines in the Elements without using slopes. The algebraic conditions appeared after analytic geometry was developed in the seventeenth century, which allowed geometric properties to be tested with equations. The relation m₁m₂ = −1 can be derived from the Pythagorean theorem.

Where are they used today?

City planners lay out grid streets with parallel and perpendicular lines. In calculus, the normal line to a curve is perpendicular to the tangent line and has the negative reciprocal slope. In computer graphics, a perpendicular direction is used to compute shadows, and in robotics, to turn at right angles. A carpenter checks a right angle with a 3-4-5 triangle, which is based on the same property.

Which mistakes are common?

Only flipping the slope

The perpendicular slope of 2 is −1/2, not 1/2 and not −2. Flip and change the sign.

Comparing intercepts

Parallel lines have different intercepts. Equal intercepts do not matter, only the slopes.

Not rewriting to slope-intercept form

In 4x − 2y = 7 the slope is 2, not 4. Solve for y first.

Using the rule with vertical lines

Treat vertical and horizontal lines separately.