Chapter 6.6 — Exercise 6.5 — Lines Parallel to Axes

Two Lines, Two Rules

Following the same table down to k = 0 specifically: x = 0 turns out to be nothing other than the y-axis itself, and likewise y = 0 turns out to be exactly the x-axis itself — the two special cases where the "distance away" shrinks to zero and the line coincides exactly with the axis it would otherwise merely run parallel to. Every other possible value of k besides zero, whether positive or negative, keeps the line strictly parallel and strictly separate from its axis — x = 5 and x = −5 sit on opposite sides of the y-axis but are both still 5 units away from it, never touching.

Rearranging Before Reading Off the Line

Most equations handed over in this exercise don't arrive already in the clean x = k or y = k shape — a small amount of ordinary rearranging almost always comes first, before the "parallel to which axis" rule can even be applied.

The rearranging step itself is nothing more than ordinary one-variable algebra, exactly the same kind practiced since long before this chapter began — 2x − 9 = 0 solves to x = 9/2 exactly as it would if y had never entered the discussion at all. What changes is only the interpretation afterward: that single solved value becomes the constant in a two-variable equation describing a whole line, not just a point on a number line. It's genuinely the same number either way, computed by the exact same steps — the only thing that shifts is what gets drawn once the number is found: a dot marked on a one-dimensional number line, or an entire vertical line drawn across a two-dimensional plane.

Writing the Equation From a Description

Going in the other direction entirely — starting from a plain-language description and working toward an equation, rather than the reverse — uses those same two rules directly, with no new ideas required at all. The line parallel to the x-axis through (2, −5) is y = −5, since only the y-coordinate matters for this rule; the x-coordinate given in the point is entirely irrelevant to deciding which specific horizontal line is actually meant here, and could just as well have been left out of the question altogether. The line parallel to the y-axis through the point (3, 5) is x = 3, for the exact same reason running in reverse this time — only the x-coordinate matters, and the y-coordinate given could be swapped for absolutely any other number without changing the answer at all.

This particular fact is worth stating out loud and plainly, since it genuinely surprises quite a few people the very first time they encounter it: a line parallel to the x-axis through (2, −5) is exactly the same line as one parallel to the x-axis through (100, −5), or (−7, −5), or any other point sharing that same y-coordinate. The other coordinate given in the point simply doesn't affect which line results at all. This is really the same fact from the "distance from an axis" table, just restated in point-form instead of distance-form: since y = b is parallel to the x-axis and sits exactly |b| units from it, every point with that same y-value — regardless of its x-value — has to lie on that identical line, because "distance from the x-axis" and "y-coordinate" turn out to be two different names for the exact same underlying measurement.

Solving an Equation by Graphing It

3x + 2 = 8x − 8 rearranges to 2 + 8 = 8x − 3x, giving 5x = 10 and x = 2 — whether it's read as an ordinary one-variable equation or genuinely graphed as x = 2 in two variables. A vertical line through that single x-value is the two-variable picture; the "solving" arithmetic itself stays completely identical either way it's approached, and only what ends up getting drawn afterward, if anything, actually differs between the two readings. This is really the clearest possible demonstration that a one-variable equation was never a fundamentally different kind of object from a two-variable one — it was always the special case where one of the two coefficients happened to equal zero, hiding in plain sight since the very first balloon example back in the introduction. This same idea generalises to any equation that reduces to a single variable after simplifying, no matter how many terms it starts out with on each side: however many x's are scattered across the original statement, everything eventually collapses into one number, and that number is simultaneously a one-variable answer and the constant defining a two-variable line.

The Chapter's Full Arc

From Ajay's balloons in the introduction through standard form, solutions, graphing, real applications, and now this final special case, every single lesson in this chapter has been quietly building toward the exact same picture: any linear equation in two variables whatsoever, without a single exception across all six lessons covered, describes a straight line — sometimes tilted, sometimes flat, sometimes upright, but always, provably, a line and nothing else. This whole framework becomes the starting point for Class 10's Pair of Linear Equations, where two such lines are studied together, and where they do or don't cross turns out to answer a genuinely new kind of question. A pair of lines that are both parallel to the same single axis, whether both of the x = k variety or both of the y = k variety, never crosses at all, however far either is extended — a direct visual echo of the fact, established all the way back in the chapter on Euclid's Geometry, that two straight lines are always either parallel or intersecting, never both, and this chapter's two special families of lines make that fairly abstract statement genuinely easy to see directly, rather than just recite it from memory.