Slope-Intercept Form

y = mx + b tells you everything you need to graph a line immediately.

b is the y-intercept: plot (0, b) first.
m is the slope: from that point, rise and run to a second point.

For y = (2/3)x − 4: start at (0, −4), then go up 2 and right 3 to reach (3, −2). Draw the line.

For a negative slope like −3, treat it as −3/1: down 3, right 1.

If an equation is not in this form, rearrange to solve for y first. 2x + 3y = 9 becomes y = (−2/3)x + 3.

Every non-vertical line has a unique representation y = mx + b, where m is the slope and (0, b) is the y-intercept. The form is the explicit expression of y as a linear function of x, which is why it is the default for graphing, for identifying parallel and perpendicular relationships, and for entering an equation into technology.

In modelling, b is the initial value and m the constant rate of change, so a phone plan costing $30 plus $0.10 per minute is C(t) = 0.10t + 30 with the units of m being dollars per minute.

Converting from standard form Ax + By = C gives m = −A/B and b = C/B, defined whenever B ≠ 0 — the excluded case being exactly the vertical lines this form cannot represent.

Parallel lines have equal m and different b; identical m and b means the same line. That distinction is what separates a system with no solution from one with infinitely many.