Chapter 6.2 — Exercise 6.1 — Standard Form

Rewriting Equations You're Given

The only genuinely real work in this first part is moving every single term across to one side of the equation without losing track of a sign anywhere along the way — 93x = 12 − 15y becomes 93x + 15y − 12 = 0, not 93x − 15y − 12 = 0, because the −15y on the right becomes +15y once it crosses to the left. The general rule behind that flip is the same one used in solving any equation: moving a term across the equals sign always reverses its sign, whether that particular term started out positive or negative, and regardless of whether it's carrying a variable along with it or just sitting there as a bare constant number.

When a Variable Is Missing Entirely

Question 2 pushes further still, with equations that only ever mention one of the two letters throughout, never both together. 2x = 5 has no y anywhere, yet it still fits the general form: 2x + 0y − 5 = 0, so b = 0 exactly. y − 2 = 0 has no x, giving a = 0: 0x + y − 2 = 0. Even y/7 = 3 works the same way, rewriting to 0x + (1/7)y − 3 = 0 — a fractional coefficient on y, a zero coefficient on x, and everything still fits neatly inside the one general template.

This is the introduction's own rule about missing variables now put directly into hands-on practice, across a whole run of six sub-questions rather than just the single passing example seen before: a "missing" letter never means the general form breaks down, only that the coefficient sitting in front of it happens to be zero. It's worth noticing that a one-variable-looking equation like 2x = 5 doesn't stop being a genuine two-variable equation just because y never appears — every value of y still satisfies it equally well once x is fixed at 5/2, since y's coefficient contributes absolutely nothing to the equation's outcome at all, regardless of what value y is actually given. That fact turns out to matter directly once graphing enters the picture: an equation with a missing variable still describes a full line on the plane, not a single isolated point — a fact that stays easy to miss from the algebra alone, and only becomes visually obvious once such an equation is actually graphed.

Turning Sentences Into Equations

Question 3 is the harder skill — building the equation from scratch rather than rearranging one already written down.

The pencil-and-pen statement is worth working through explicitly, since it's the only one of the five mixing two genuinely different unit prices together into a single combined total rather than relating two quantities of the same kind. Letting x be the number of pencils and y the number of pens: pencils cost 2x rupees in total, pens cost 15y rupees in total, and the combined bill is 100 — giving 2x + 15y = 100 directly, no different in structure from Hari's pens-and-pencils example in the introduction, just with quantities as the unknowns instead of prices. The habit worth building from this question is deciding what x and y actually stand for before writing anything else down — every one of these five statements becomes almost mechanical to translate once the two unknown quantities are named clearly and consistently, but genuinely difficult to get right if that basic naming step gets skipped over, rushed, or left only implicit somewhere in the working.

A Two-Digit Number in Disguise

The single trickiest translation in the whole exercise involves a two-digit number together with its own digit-reversal. With x as the unit's digit and y as the ten's digit, the original number is 10y + x — not simply the two digits "xy" written side by side as separate symbols on the page, which is really the exact mistake this particular question is testing for — since the ten's digit contributes ten times its own face value to the number's actual size, not merely one, the way each digit alone contributes on its own. Reversing the digits swaps which one plays which role, giving 10x + y for the reversed number — the same two digits, just with the multiplier of 10 now attached to the other one. Their sum: (10y + x) + (10x + y) = 11x + 11y = 11(x + y). Told that this sum is 121, dividing both sides by 11 collapses the whole equation down to x + y = 11 — a much simpler relationship than the original wording suggested. This is a case worth remembering as a pattern in its own right: whenever a two-digit number and its digit-reversal both get added together, the ten's and unit's contributions from each number pair up and cancel out most of their individual weighting, leaving a clean multiple of 11 behind every single time, regardless of which two specific digits were chosen to begin with — which is exactly why the problem's answer collapses down to the strikingly clean x + y = 11 rather than staying tangled up in the original 121.

Carrying the Translation Skill Forward

Writing an equation is only step one — Exercise 6.2 picks up exactly where this leaves off, asking what it actually means for a pair of numbers to satisfy an equation once it's been written, and how many such pairs exist for any single one of them. An equation like the pencil-and-pen one written here stops being just a translated statement sitting on the page and starts being actively tested against specific candidate number-pairs to see which ones genuinely satisfy it.