5.3 Slope and Parallel/Perpendicular Lines

Slope and Parallel/Perpendicular Lines

Quick Answer: Slope measures the steepness and direction of a line. On the Florida Algebra 1 EOC, MA.912.AR.2.3 tests your ability to calculate slope from two points, a table, or a graph, and to use slope relationships to identify, write, and graph equations of parallel and perpendicular lines.

What Is Slope?

Slope (m) is the ratio of vertical change to horizontal change between two points on a line — "rise over run":

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are any two distinct points on the line.

• m > 0: line increases left to right (upward).
• m < 0: line decreases left to right (downward).
• m = 0: horizontal line.
• m undefined: vertical line.

Calculating Slope from Different Representations

From two points: Given (2, 5) and (6, 13): m = (13 - 5) / (6 - 2) = 8/4 = 2.

From a table: If x-values change by a constant increment and y-values change by a constant increment, the ratio of the y-change to the x-change is the slope. Pick any two rows.

• Rows (0, 3) and (2, 7): m = (7 - 3) / (2 - 0) = 4/2 = 2.

From a graph: Count the vertical change (rise) and horizontal change (run) between two lattice points. Up 3 and right 5 gives m = 3/5. Down 4 and right 2 gives m = -2.

Exam trap: The slope formula is NOT commutative in the middle. If you subtract y1 - y2 in the numerator, you must subtract x1 - x2 in the denominator — never mix directions. (13 - 5) / (6 - 2) and (5 - 13) / (2 - 6) both equal 2, but (13 - 5) / (2 - 6) equals -2, a sign-flipped trap.

Parallel Lines: Equal Slopes

Two non-vertical lines are parallel if and only if they have the same slope and different y-intercepts. Same slope and same y-intercept means the same line (coincident), not parallel.

All vertical lines (e.g., x = 2 and x = -5) are parallel to each other, with undefined slope.

Perpendicular Lines: Negative Reciprocals

Two non-vertical lines are perpendicular if and only if their slopes are negative reciprocals of each other. The product of perpendicular slopes is always -1:

m1 * m2 = -1

• If m1 = 2, then m2 = -1/2.
• If m1 = -3/4, then m2 = 4/3.
• If m1 = 1, then m2 = -1.

A horizontal line (m = 0) is perpendicular to a vertical line (undefined slope).

Writing Equations of Parallel and Perpendicular Lines

Florida items often give a line and a point, then ask for the equation of a parallel or perpendicular line through that point.

Procedure:

1. Identify the slope m1 of the given line (rewrite in slope-intercept form if needed).
2. Determine the required slope: parallel uses m = m1; perpendicular uses m = -1/m1 (flip the fraction and change the sign).
3. Substitute the given point (x0, y0) and the new slope into point-slope form: y - y0 = m(x - x0).
4. Convert to slope-intercept form if the item requires it.

Worked example (parallel): Line parallel to y = -2x + 5 through (3, 4).

• Parallel slope: m = -2. Point-slope: y - 4 = -2(x - 3) → y = -2x + 10.

Worked example (perpendicular): Line perpendicular to y = (3/4)x + 2 through (6, -1).

• Perpendicular slope: m = -4/3. Point-slope: y + 1 = (-4/3)(x - 6) → y = (-4/3)x + 7.

Exam Traps

1. Confusing perpendicular with opposite: m1 = 3 does NOT have perpendicular slope -3. The perpendicular slope is -1/3 — you must also flip the fraction.
2. Forgetting to flip the fraction: If m1 = 2/5, the perpendicular slope is -5/2, not -2/5.
3. Sign error in slope formula: Subtracting in the wrong order flips the slope sign. Be consistent.
4. Treating undefined slope incorrectly: A vertical line has undefined slope, not 0. A horizontal line has slope 0, not undefined.
5. Misidentifying parallel as coincident: Same slope and same intercept means the same line, not parallel.
6. Using the wrong point in point-slope form: Use the given point for the new line, not a point from the original line.

Real-World Context

Florida items embed perpendicularity in geometric context. Two roads meet at a right angle. If one road has equation y = (2/3)x - 4, the perpendicular road has slope -3/2. Through (3, -2): y + 2 = (-3/2)(x - 3) → y = (-3/2)x + 5/2. This appears in staircase rails, roof pitches, and maps.