1. What makes an equation linear
An equation in which every variable appears only to the first power, with no products of variables and no functions of variables (no squares, no roots, no $\sin$, no $\log$, no $1/x$).
The defining trait is structural, not visual. An equation is linear if, when you collect terms, every variable shows up like $x$ — never like $x^2$, never like $xy$, never inside a function. Constants and constant coefficients are free to do whatever they want.
Examples and non-examples
- $3x + 7 = 19$
- $2x + 3y = 12$
- $y = -\tfrac{1}{2}x + 4$
- $x + y + z = 0$
- $x^2 + 3 = 7$ — squared variable
- $xy = 6$ — product of variables
- $\sqrt{x} = 4$ — root
- $\tfrac{1}{x} + 2 = 5$ — variable in denominator
The name comes from the picture: every equation that fits the rule above traces out a line when you graph it. Two variables give a line in 2-D; three give a flat plane in 3-D; $n$ give what mathematicians call a hyperplane. The algebra and the geometry are the same thing seen from two angles.
2. Solving a one-variable linear equation
Every one-variable linear equation can be rearranged into the canonical form
$$ ax + b = 0 $$where $a$ and $b$ are constants and $a \neq 0$. Solving it means finding the single value of $x$ that makes the equation true. The recipe is one rule applied twice:
Whatever operation you do, do it to both sides.
Subtract $b$ from both sides:
$$ ax = -b $$Divide both sides by $a$:
$$ x = -\frac{b}{a} $$That's it. Every one-variable linear equation collapses into two moves: isolate the term with $x$, then divide by the coefficient of $x$.
The shape of the recipe
The recipe quietly assumes $a \neq 0$. When $a = 0$, the equation reduces to $b = 0$ — which is either trivially true (for every $x$) if $b$ really is zero, or impossible if it isn't. Either way, there is no single $x$ to solve for. Always check the coefficient before dividing.
When the equation has more than two steps
Real equations don't always arrive in the tidy form $ax + b = 0$. Two patterns come up constantly, and both reduce to the same recipe once you set them up.
Distribute, then solve
When the variable is trapped inside parentheses, multiply through first:
$$ 3(x + 4) = 21 $$Distribute the $3$:
$$ 3x + 12 = 21 $$Now it's an ordinary two-step equation. Subtract $12$, divide by $3$:
$$ 3x = 9 \quad\Longrightarrow\quad x = 3 $$Variables on both sides
When $x$ appears on the left and the right, collect the $x$ terms on one side and the constants on the other. Choose whichever side keeps the $x$ coefficient positive — life is easier that way:
$$ 2x + 3 = 5x - 9 $$Subtract $2x$ from both sides (smaller coefficient leaves the equation cleaner):
$$ 3 = 3x - 9 $$Add $9$ to both sides, then divide by $3$:
$$ 12 = 3x \quad\Longrightarrow\quad x = 4 $$The principle hasn't changed — you're still doing the same operation to both sides to isolate $x$. There are just more setup moves before the final divide.
3. Two variables: a line in the plane
The moment you go from one variable to two, the answer changes shape. A single number isn't enough anymore — you need a pair $(x, y)$. And there isn't just one solution; there are infinitely many. Together they form a line.
Consider $2x + 3y = 12$. Try a few values:
| $x$ | $y$ | Check $2x + 3y$ |
|---|---|---|
| $0$ | $4$ | $0 + 12 = 12$ ✓ |
| $3$ | $2$ | $6 + 6 = 12$ ✓ |
| $6$ | $0$ | $12 + 0 = 12$ ✓ |
| $-3$ | $6$ | $-6 + 18 = 12$ ✓ |
Every one of those pairs satisfies the equation, and if you plot them on a grid they fall in a perfectly straight row. The equation isn't a riddle with one answer — it's a constraint, and the line is the set of every $(x, y)$ that obeys it.
A linear equation in two variables is a rule that throws away most pairs $(x, y)$ in the plane and keeps only the ones that lie on a particular line. The line is the equation.
4. Three forms of the same line
The same line can be written in several ways. Each form makes a different fact about the line immediately readable, and you pick the form that surfaces what you need.
| Form | Equation | Reads off easily | Best when |
|---|---|---|---|
| Slope-intercept | $y = mx + b$ | Slope $m$, y-intercept $b$ | Graphing quickly; comparing two lines' steepness |
| Point-slope | $y - y_1 = m(x - x_1)$ | Slope $m$, a point $(x_1, y_1)$ | You know a point and a slope and want the equation |
| Standard | $Ax + By = C$ | Integer coefficients; symmetry in $x,y$ | Solving systems; clean algebra |
They are interchangeable. Rearranging is just algebra, and you should be able to flip between them without thinking. For example, $2x + 3y = 12$ (standard) becomes $y = -\tfrac{2}{3}x + 4$ (slope-intercept) by solving for $y$.
5. Slope and intercepts
Slope is the line's steepness — how much $y$ changes for each one-unit step in $x$. Given any two points on the line:
$$ m = \frac{y_2 - y_1}{x_2 - x_1} \quad \text{(rise over run)} $$A positive slope rises left-to-right; negative falls; zero is horizontal; undefined (division by zero) is vertical.
The y-intercept is where the line crosses the y-axis — set $x = 0$ and solve for $y$. In slope-intercept form $y = mx + b$, it's just $b$, sitting there for free.
The x-intercept is the mirror: where the line crosses the x-axis. Set $y = 0$ and solve for $x$. In slope-intercept form that gives $x = -b/m$ (when $m \neq 0$).
Reading intercepts from standard form
When the line is given in standard form $Ax + By = C$, you don't need to convert to slope-intercept just to find an intercept. Use the definitions directly:
- y-intercept: set $x = 0$ and solve for $y$.
- x-intercept: set $y = 0$ and solve for $x$.
Example: for $2x - 3y = 12$, setting $x = 0$ gives $-3y = 12$, so the y-intercept is $y = -4$. Setting $y = 0$ gives $2x = 12$, so the x-intercept is $x = 6$. Two points, one line — that's already enough to graph it.
Parallel and perpendicular slopes
Two slope relationships show up so often that you should know them cold.
Two non-vertical lines are parallel if and only if they have the same slope. If $m_1 = m_2$ and the y-intercepts differ, the lines run alongside each other and never meet.
Two non-vertical lines are perpendicular (meeting at a right angle) if and only if their slopes are negative reciprocals: $m_1 \cdot m_2 = -1$, equivalently $m_2 = -\dfrac{1}{m_1}$.
So a line parallel to $y = 3x - 1$ has slope $3$ (anything else fails). A line perpendicular to it has slope $-\tfrac{1}{3}$ (check: $3 \cdot -\tfrac{1}{3} = -1$).
The perpendicular formula breaks down at the boundary. A horizontal line has slope $0$, and its perpendicular is a vertical line — which has undefined slope, not $-\tfrac{1}{0}$. When one slope is $0$ or undefined, don't use the negative-reciprocal rule; just remember that horizontals and verticals are perpendicular to each other.