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Algebra foundations · 03 · Straight lines · 8 min

Linear relationships

When something changes at a steady rate — a fee per kilometer, a fill rate per minute — its graph is a straight line, and two numbers tell its whole story: where it starts, and how fast it climbs.

Build the intuition

y = mx + b, decoded

b is where the line starts (the value when x = 0 — the base fee, the empty tank). m is the slope: how much y changes per unit of x (the per-kilometer rate). Read any linear formula this way and it turns from symbols into a sentence.

y=mx+by = mx + b

Slope is a rate with units

m = 2 doesn't just mean “slanted” — it means 2 dollars per kilometer, 2 liters per minute, 2 degrees per hour. Naming the units makes slopes meaningful and mistakes obvious.

Parallel stories, one intersection

Two linear situations — two phone plans, two rental cars — cross where their lines intersect. Before the crossing one is cheaper; after it, the other. Finding the break-even point is solving mx + b = nx + c, and it settles real arguments.

See it move

Interactivey = mx + b, in your hands
1.5
-1
y = 1.5x − 1: starts at height -1, then rises 1.5 for every step right. Two numbers, the whole story.

Two dials, every line: b lifts the start, m sets the climb. Watch the slope triangle keep count.

A worked example

Which gym wins?

  1. Gym A: $50 to join, $10/month. Gym B: no join fee, $15/month.

  2. Model both:

    A=10m+50,B=15mA = 10m + 50, \quad B = 15m

  3. Break-even:

    10m+50=15m  ⇒  m=1010m + 50 = 15m \;\Rightarrow\; m = 10

  4. Under 10 months, B is cheaper; beyond 10, A wins. The intersection answered the question.

Out in the world

Trend lines everywhere

Sales projections, climate trends, fitness progress — the first model anyone fits to data is a straight line. “Roughly linear” is the most useful approximation in all of quantitative work.

Common confusion, cleared

“Slope is how high the line is.”

Slope is how fast it rises — a line can start high and climb slowly, or start low and climb fast. Height is b's department; climb is m's.

“A negative slope means something went wrong.”

It means the quantity decreases as x grows — battery draining, debt shrinking. Downhill lines model half of life.

Recap

  • Linear = steady rate of change = straight line.
  • b is the start; m is the rate, with units.
  • Intersections of lines answer break-even questions.