Linear Equations in Two Variables

Every month of phone bills, drawn at once

The phone-bill machine from F8 — $$20$ plus $$3$ per gigabyte — has been following you through algebra. Plot it. A month of $1$ GB costs $$23$: the point $(1, 23)$. Four gigabytes, $(4, 32)$. Every possible month lands on one perfectly straight line, and that line has an equation:

$y = 3x + 20$

Read it with A3 eyes: the $3$ is the slope — the bill climbs $$3$ per gigabyte of run — and the $20$ is the y-intercept, the fixed fee waiting at zero usage, the point $(0, 20)$. That’s slope-intercept form, $y = mx + b$: know a line’s tilt and one anchor point on the y-axis and you know the whole line.

What does it mean for a point to be “on” the line? Just that its address satisfies the equation. Is $(4, 32)$ on this line? Check: $3(4) + 20 = 32$ ✓. Is $(5, 36)$? $3(5) + 20 = 35 \ne 36$ — that month doesn’t exist on this plan. An equation in two variables is a membership test for points.

Watch the two numbers work one at a time — $b$ slides the line, then $m$ tilts it around its anchor:

Now take the controls yourself:

Drag the numbers: predict first, then check. Make $m$ negative — the line falls to the right. Set $m = 0$ — A3’s flat line, the “slope zero” case. Then leave $m$ alone and slide $b$: the tilt never changes, the whole line just rides up and down.

One line, three outfits

The same line can be dressed three ways, and you pick the outfit that matches the information you’re handed:

Point-slope form looks alien until you see where it comes from. Take A3’s slope formula between a known point $(x_1, y_1)$ and any other point $(x, y)$ on the line: $m = \frac{y - y_1}{x - x_1}$. Multiply both sides by the run, and there it is: $y - y_1 = m(x - x_1)$. It isn’t a new formula to memorize — it’s the definition of slope, rearranged. Use it whenever you know the slope and one point that isn’t the y-intercept: plug in, done.

Standard form $Ax + By = C$ (integers, usually $A \ge 0$) is the natural shape of combined purchase problems: $3$ burgers and $2$ fries for $$18$ is $3x + 2y = 18$ before you’ve done any algebra at all. The SAT leans on this form constantly, precisely because nothing in it announces the slope.

Reading a line out of standard form

To see the slope hiding in $2x - 3y = 6$, do the honest thing: solve for $y$ with A1 moves. Subtract $2x$: $-3y = -2x + 6$. Divide by $-3$: $y = \tfrac{2}{3}x - 2$. Same line, slope now visible. The shortcut formulas —

$\text{slope} = -\frac{A}{B}, \qquad \text{y-intercept} = \frac{C}{B}, \qquad \text{x-intercept} = \frac{C}{A}$

— are that division done once in general, and the minus sign in $-\frac{A}{B}$ is exactly the “divide by $B$‘s sign” step people skip when they eyeball it. If you remember the shortcut, remember its minus; if you doubt the minus, just solve for $y$ — thirty seconds, no faith required.

One more eyeball trap while we’re here: in $y = 4 - 3x$, the slope is $-3$, not $4$. Years of $y = mx + b$ examples put the slope first after the equals sign, so the eye grabs the $4$. The real rule has no exceptions: the slope is whatever multiplies $x$ (sign included); the lone constant is the intercept — wherever each happens to sit.

Here is that very line, live. Switch its costume and watch the plane: the algebra rewrites itself, the line never moves a pixel — that is what “the same line” means.

Parallel & perpendicular

Parallel lines never meet because they climb at identical rates: same slope, different y-intercepts. (Same slope and same intercept isn’t parallel — it’s the same line twice.)

Perpendicular is a quarter-turn, and the quarter-turn does something neat to the slope triangle: what used to be the run becomes the rise, and one leg changes direction. Rotate the “over $3$, up $2$” triangle ($m = \tfrac{2}{3}$) by $90°$ and it becomes “over $-2$, up $3$” ($m = -\tfrac{3}{2}$): the fraction flips and the sign negates. Do both, always — flip alone or negate alone gives a line that’s merely different, not perpendicular.

A special pair sits outside the formula: a horizontal line (slope $0$) and a vertical line (undefined slope) are perpendicular, even though $0$ has no reciprocal to flip.

From two points to the whole equation

Real problems often hand you nothing but two points. The pipeline runs entirely on tools you have: slope from A3’s formula, then point-slope with either point, then tidy into slope-intercept. Feed the From Two Points tab the staircase points from A3 — $(2, 1)$ and $(6, 9)$ — and predict the slope and intercept before it builds the equation.

The one thing to remember

A two-variable equation is a membership test for points, and a line is everything that passes. $y = mx + b$ shows the rate and the starting value; point-slope is the slope formula rearranged; standard form hides the slope at $-\frac{A}{B}$ until you solve for $y$. Parallel means same slope; perpendicular means flip and negate.

The three forms

From standard form $Ax + By = C$:

• slope $= -\dfrac{A}{B}$
• y-intercept $= \dfrac{C}{B}$,   x-intercept $= \dfrac{C}{A}$

Example: $2x - 3y = 6 \Rightarrow$ slope $= -\tfrac{2}{-3} = \tfrac{2}{3}$,   y-int $= \tfrac{6}{-3} = -2$.

Parallel & perpendicular