L4 · Parallel and perpendicular slopes; why a linear function has exactly one x-intercept

Learning goal

“I can compare lines by their slopes - equal slopes (with different intercepts) mean parallel, and slopes that are negative reciprocals mean perpendicular - and I can prove that a linear function with slope m != 0 has exactly one x-intercept.”

Parallel lines — lines in a plane that never meet; for graphs, they have equal slopes and different y-intercepts.

Perpendicular lines — lines that meet at a right angle (90 degrees).

Negative reciprocal — of a slope m: flip the fraction and change the sign. The negative reciprocal of 3 is -1/3; of -2/5 it is 5/2. Perpendicular slopes are negative reciprocals of each other, and their product is -1.

Exactly one solution — an equation like mx = -b (m != 0) has a single value of x that works; this is the engine of the x-intercept proof in this lesson.

Worked example

Line 1 is y = 2x + 1. Line 2 is described by two points: (0, -3) and (2, 1). Step 1: Line 2's slope: rise = 1 - (-3) = 4, run = 2, slope = 2. Step 2: The slopes are equal (2 and 2) and the y-intercepts differ (1 vs -3), so the lines climb identically and start at different heights - they can never meet. They are parallel. Now turn one line: the line y = -(1/2)x + 4 has slope -1/2. Compare with slope 2: -1/2 is the negative reciprocal of 2 (flip 2/1 to 1/2, change the sign), and 2 x (-1/2) = -1. Lines whose slopes multiply to -1 meet at a right angle: they are perpendicular. Say it plainly: slope comparison is a complete test. Equal -> parallel. Negative reciprocals -> perpendicular. Anything else -> the lines meet at some ordinary angle.

Check yourself — explain it

Slopes 3 and -3 are NOT perpendicular because...

Paired problem — example still visible

Are the lines y = 3x - 2 and the line through (0, 5) and (1, 8) parallel? -> Yes - the second line's slope is (8 - 5)/(1 - 0) = 3, equal to the first line's slope, and the intercepts differ (5 vs -2).

Faded completion — last step is yours

"A line perpendicular to y = 5x + 1 has slope ___" -> -1/5

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10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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