Linear equation

How to recognise one

Every variable is raised to the first power and nothing sits in a denominator:

Linear	Not linear	Why not
$y = 2x + 3$	$y = x^2 + 3$	squared term
$3x + 2y = 12$	$xy = 6$	variables multiplied together
$y = -\tfrac{1}{2}x$	$y = \tfrac{1}{x}$	variable in a denominator
$x = 4$	$y = \sqrt{x}$	variable under a root

A fraction is fine as long as it is a coefficient, as in $\tfrac{1}{2}x$. The problem is a variable in the denominator.

Three forms of the same line

Form	Looks like	Best for
Slope-intercept	$y = mx + b$	reading slope and intercept, and graphing quickly
Standard	$Ax + By = C$	finding both intercepts
Point-slope	$y - y_1 = m(x - x_1)$	building an equation from a point and a slope

They describe the same line and convert into one another by rearranging — the same skill as literal equations.

Why the graph is straight

A linear equation has a constant slope: moving one unit across always changes $y$ by the same amount. Constant steepness is exactly what “straight” means. A quadratic has a steepness that changes as you move along it, which is why it curves.