Chapter 6.1 — Introduction to Linear Equations (2 Variables)

When One Unknown Isn't Enough

Hari bought 5 pens and 4 pencils for ₹32, and neither price is known individually. Calling the pen's unknown price x and the pencil's unknown price y: 5 pens cost 5x rupees, 4 pencils cost 4y rupees, and together the two amounts total 32 — giving 5x + 4y = 32, one single equation carrying two genuinely separate unknown prices at once, rather than just one as before. Both variables have exponent 1, so this is a linear equation in two variables. p + 2q = 15, 2.5l − 3.5m = 1, and a/5 − b/6 = 1 are further examples — different letter pairs each time, but always exactly the same underlying shape underneath. What makes each of these "linear" is exactly what made 5x = 40 linear: no variable is ever squared, multiplied by another variable, or buried underneath a root sign — every term is just a plain number times a single variable raised to the first power, or else a bare constant sitting on its own.

Rewriting Any Equation Into Standard Form

Almost any linear equation in two variables can be rearranged into ax + by + c = 0 by moving every term to one side. The value of doing this is that a, b, and c become instantly readable once the rearranging is done, without needing to solve or simplify anything further — the standard form is never a genuinely different equation from the one originally given, only that same equation relabelled so its three defining numbers are sitting in plain sight, ready to be read off directly rather than picked out from wherever they happened to land in the original statement.

Working through the fourth row shows the mechanics clearly: x/2 − y/3 − 5 = 0 already has everything on one side, so it just needs its fractional coefficients written out as ½ and −⅓ rather than left as division. The last row is worth pausing on separately: 2x = y has no visible constant term at all, yet c is still exactly 0 rather than simply absent — the general form always has a slot for c, and "no constant" just means that slot is filled with zero rather than left empty. The same habit applies to a missing variable, not just a missing constant: an equation like y = 8, with no x term visible anywhere, still fits the general form with a = 0 — 0x + y − 8 = 0 — the coefficient of the missing variable is 0, not something the general form can't handle. The same logic runs in reverse for reading a, b, and c back off a given equation: before naming any of the three values, it's worth checking explicitly whether both x and y actually appear, because a missing letter always means a zero coefficient for that letter, silently sitting in the general form even though nothing was written down for it.

Why a and b Can't Both Be Zero

If a and b were both 0, the equation would collapse to c = 0 — nothing more than a bare statement about a single number, with no trace of x or y left in it anywhere. That's not a linear equation in two variables anymore, just a true-or-false claim about a constant, which is exactly why the general form explicitly rules out a and b both being zero at the exact same time. Either coefficient alone is still free to be zero — a = 0 together with c = −5 gives 0x + by + c = 0, which simplifies to just by + c = 0 — just never both a and b together. It's worth seeing why this restriction matters rather than just accepting it as a rule: the whole point of the general form is to describe a relationship between x and y, and an equation with neither variable present describes no relationship between them at all — it's either always true or always false, regardless of what x and y happen to be, which makes it a completely different kind of statement from every other genuine linear equation studied throughout the rest of this chapter. Every other row in the table above, by contrast, is a real constraint linking x and y together — pick any value for one and the equation pins down exactly what the other must be, which is precisely the relationship a and b being not-both-zero is there to guarantee stays possible.

What Fractional Coefficients Mean

The row x/2 − y/3 − 5 = 0 shows that a, b, and c don't have to be whole numbers. Converting x/2 to ½x and −y/3 to −⅓y is the same move used throughout ordinary algebra — a coefficient is just whatever multiplies the variable, whether that's a whole number, a fraction, or a decimal like 2.5, as in 2.5l − 3.5m = 1 from the examples above. The reason it's worth writing 1/2 as ½ (or equivalently as 0.5) explicitly, rather than leaving x/2 in its original division form, is that the general form ax + by + c = 0 is defined around multiplication — a, b, and c are coefficients being multiplied by the variables, and "x divided by 2" is really just shorthand for "x multiplied by one-half," wearing a slightly different, but entirely equivalent, notation on the page.

Where This Groundwork Leads

Every exercise in this chapter builds directly on the general form fixed here. Exercise 6.1 drills the standard-form conversion itself, including several trickier cases with a missing variable term; later exercises move on to what a "solution" of such an equation even means, and what its graph looks like. That last question turns out to have a genuinely different flavour from anything a one-variable equation offered: 5x = 40 had exactly one solution, a single number, but an equation in two variables — with a whole extra degree of freedom to work with — will turn out to have infinitely many, every one of them a coordinate pair rather than a single lone number the way 5x = 40's solution was.