The Line, Written Down
In y = mx + b, the intercept b and the slope m carry the whole line — enough to graph it at sight, or to write it from two points. · 12 min
A line runs forever, yet you can hand it to a friend in a few characters: y = 2x + 1. This folio is about that compression. Two numbers carry everything — where the line starts on the y-axis, and the rate it keeps from there. Learn to read them fluently and you can draw any line straight from its equation, and write the equation of any line you are handed.
Guess before you learn
The line y = 2x + 5. What is y when x = 0?
At x = 0 the term 2x vanishes, leaving y = 5 — so the line crosses the y-axis at (0, 5). That number has a name, an address, and a letter reserved for it: b. If you said 7, you computed x = 1 — a natural slip, and worth one careful pencil mark.
9–12
3–5
A plant is 3 cm tall and grows 2 cm each week, so its height is 3 + 2 × weeks. The 3 is the starting height; the 2 is the growth per week. Change either number and you describe a different plant — but always the same kind of story: start somewhere, grow steadily.
On a graph, the starting number shows where the line meets the upright axis, and the growth number tilts the line.
6–8
In y = mx + b, each letter earns its place. Set x = 0 and y = b: the line crosses the y-axis at (0, b), so b is the y-intercept — the starting value. The coefficient m is the slope from last folio: from (0, b), every step of 1 in x moves y by m.
To graph y = 3x − 2: dot the intercept (0, −2), step over 1 and up 3, step again, and rule the line through. No table of values needed — the equation is already a set of instructions.
9–12
Slope-intercept form, y = mx + b, packs a full description into two parameters: b, the value of y at x = 0, and m, the constant rate. Graphing runs forward — plot (0, b), then step the slope. Writing an equation runs backward: given the slope and any one point, substitute both into y = mx + b, and the only unknown left standing is b. Solve for it and assemble the equation.
Given two points instead, find m first with the slope formula, then recover b through either point — both give the same b, which makes a free check. One caution: a vertical line has no m, so it refuses this form entirely and is written x = a.
K–2
You have 5 blocks. Every day you add 2 more. Day 1: 7 blocks. Day 2: 9. Day 3: 11. Two numbers tell the whole story — where you start, and how many you add.
Start somewhere. Add the same amount, again and again. Every straight line is built exactly this way.
Undergrad
The map (m, b) ↦ (x ↦ mx + b) is a bijection between ℝ² and the non-vertical lines: existence and uniqueness of the line through two points with distinct x-coordinates amounts to a 2 × 2 linear system whose determinant, x₂ − x₁, is nonzero. Point-slope form, y − y₁ = m(x − x₁), is often the sharper tool — it privileges no particular point and mirrors the expansion f(x) = f(x₁) + m(x − x₁).
Recovering b is nothing more than evaluation at 0: the intercept is not extra information but a value the line already carries.
Postgrad
Lines in the plane form a two-parameter family; slope-intercept coordinates (m, b) chart that family minus the vertical pencil, which lives at infinity in the projective completion. The general form ax + by = c covers all lines at the price of a scale redundancy — a first taste of homogeneous coordinates.
Point-slope form makes the affine structure explicit: fixing a base point identifies the line's points with ℝ via the direction vector (1, m). The recurring pattern — choose the chart adapted to your data, translate between charts freely — is the actual skill this folio installs.
y-intercept
The point (0, b) where a line crosses the y-axis — the value of y when x = 0. In y = mx + b, it is the b.
Now reverse the reading. A line is pinned down by a slope and one point, or by two points — either way, the equation can be recovered. From slope and a point: you already know m, so write y = mx + b, substitute the point's coordinates for x and y, and the only letter left standing is b. Solve for it. From two points: compute m first with the slope formula, then proceed exactly as before, using either point. Both points give the same b — which makes a free check.
The line with slope 3 through (2, 7) — the steps fade as you master them
y = 3x + b
7 = 3(2) + b
7 = 6 + b
b = 1
y = 3x + 1
Why is this true?
Why does substituting a known point into y = mx + b let you find b?
A point on the line makes the equation true, so its coordinates satisfy y = mx + b. With x, y, and m all known numbers, b is the only unknown left, and one balancing step isolates it.
One line, two readable numbers. You can now travel both directions: equation to graph, and points to equation. Next folio asks a bolder question — what happens when two lines must hold at once, and what a single pair (x, y) has to do to satisfy them both.
Practice — new ink and old, interleaved
1.Put the table check in working order.
- Compare the outputs attached to any repeated input
- Scan the input column for a value that repeats
- Two different outputs for one input: not a function
2.A ramp runs from (0, 0) to (12, 3). What is its slope?
3.In y = x − 7, what are m and b?
4.y = −4x + 9. What is y when x = 2?
5.Match each equation to its line.
6.Which of these tables could come from a function?
7.From memory: how do you write the equation of a line from two points?
Find m = (y₂ − y₁)/(x₂ − x₁), substitute m and one point into y = mx + b to solve for b, then write the equation and check it with the other point.
How close were you? Grade yourself honestly — it sets your review date.