Chapter 6.4 — Exercise 6.3 — Graphical Method

Five Steps From Equation to Line

• Rewrite the equation as y = f(x), isolating y completely on one side of the equals sign, with every other term moved across.
• Substitute a few different chosen values for x in turn, and compute whatever matching value of y each substitution produces.
• Write each resulting pair down explicitly as a coordinate (x, y), ready to be located on the plane. Two points are technically enough to fix a straight line uniquely, but a third acts as a valuable built-in check — if all three don't fall neatly on one line once plotted, an arithmetic slip somewhere in the substitution step is the near-certain explanation, and it's worth re-checking before drawing anything at all.
• Plot every one of those computed points carefully onto the Cartesian plane, using the same axis-reading skill built up in the coordinate geometry chapter earlier.
• Join every plotted point in turn with a ruler — the result, provided every single point was placed correctly beforehand, is always a perfectly straight line, never a slight curve or kink anywhere along it.

Working through 3x + y = 7 as a concrete case, rewritten first as y = 7 − 3x: x = 1 gives y = 4, x = 3 gives y = −2, x = 0 gives y = 7. Three points, (1, 4), (3, −2), (0, 7), and all three land on one line when plotted. Choosing which three x-values to substitute is entirely free — nothing about the underlying method requires the specific values 0, 1, and 3 in particular, only that the resulting points spread out enough to draw the line confidently and accurately rather than being crammed too close together to see a clear direction.

Reading Values Off an Already-Drawn Graph

Once a line is fully drawn, it answers questions that no single isolated solution ever could on its own. A table of three solutions only speaks to those three specific x-values; the completed line, by contrast, holds an answer ready for literally every x-value along its length, whether or not that particular value was ever explicitly calculated ahead of time. For y − x = 2, plotted through (1, 3), (3, 5), and (−1, 1): reading the graph at x = 4 gives y = 6, and reading it at y = −3 gives x = −5 — both consistent with the equation itself (6 − 4 = 2, and −3 − (−5) = 2), just found by tracing along the already-drawn line instead of solving the equation freshly, purely algebraically, each time a new value is needed. This is really the entire point of drawing the graph in the first place — once the line exists on paper, any question of the form "what's y when x is such-and-such" or "what's x when y is such-and-such" becomes a matter of finding one axis value and reading straight across or straight up to where the line happens to sit, rather than resolving the original equation freshly for every single question asked.

The check against the algebra is worth doing every time a value is read off a graph, not just occasionally. Substituting the read-off pair back into the original equation — 6 − 4 = 2 for (4, 6), or −3 − (−5) = 2 for (−5, −3) — confirms the graph was read accurately, and catches the kind of small misreading that's genuinely easy to make whenever a point sits somewhere between two grid lines rather than falling exactly on one of them.

What Every Line y = mx Has in Common

Plotting y = x, y = 2x, y = −2x, y = 3x, and y = −3x together reveals a shared property: every single one passes through the origin, (0, 0). That's not specific to these five slopes — substituting x = 0 into y = mx always gives y = 0, for any value of m at all, so a missing constant term (c = 0 in general form) guarantees the line passes through the origin before a single point is even plotted. The slope itself, the value of m, controls only how steeply the line rises or falls and in which direction — y = 3x climbs three times as fast as y = x for every step to the right, and the negative-slope pair, y = −2x and y = −3x, fall instead of rise, tilting down and to the right rather than up. But every one of them, steep or shallow, rising or falling, still shares that single fixed anchor point at the origin, since none of them carries a constant term to shift it away.

Where a Line Crosses the Axes

A line's intersection with each of the two coordinate axes is itself a detail worth finding directly and deliberately, rather than only stumbling across it by chance while plotting other points. For 6x − 3y = 12, rewritten as y = 2(x − 2): setting y = 0 gives x = 2, so the line crosses the x-axis at (2, 0); setting x = 0 gives y = −4, so it crosses the y-axis at (0, −4) instead. These two crossing points alone, with no third point needed, are already enough to draw the entire line — a shortcut worth knowing alongside the standard three-points method. These two crossing points have their own names, worth remembering: the point where a line meets the x-axis is its x-intercept, and where it meets the y-axis is its y-intercept. Finding both is often faster than the general three-point method precisely because each one only ever requires solving a one-variable equation — set y to 0, or set x to 0 instead, and whichever variable remains drops out of the picture entirely for that one particular calculation.

From Drawing Lines to Solving Real Problems

Every skill fixed here — rewriting as y = f(x), plotting, reading values back off a finished graph — is exactly what Exercise 6.4 immediately puts to use, this time on equations that come from real situations rather than bare algebra, with the graph itself used to answer the original question. In every such problem, the underlying equation still needs deriving from the words first — the graphing technique built up over the course of this exercise only takes over once that earlier translation step, practiced back in Exercise 6.1, has already produced something in the familiar y = f(x) or ax + by + c = 0 shape, ready and waiting to be plotted point by point.