Gradient and intercepts

A straight-line equation is an algebraic relation between two variables that graphs as a line with constant gradient. The gradient measures the change in the vertical coordinate relative to the change in the horizontal coordinate:

A positive gradient rises from left to right, while a negative gradient falls. A zero gradient produces a horizontal line. For a vertical line, the gradient is undefined because the horizontal change is zero.

An intercept is a point where a graph meets a coordinate axis. At an $x$-intercept, $y=0$; at a $y$-intercept, $x=0$. To find either one algebraically, substitute zero for the other coordinate instead of reading an approximate value from the graph.

Forms of a straight-line equation

The gradient-intercept form shows the gradient and vertical intercept directly:

The line crosses the $y$-axis at $(0,c)$. This form is usually the quickest to interpret or graph.

The general form collects all terms on one side:

Their units must make the terms compatible; in a unitless coordinate system, they are dimensionless. When $b\ne0$, rearrangement gives

so the gradient is $-a/b$ and the $y$-intercept is $-d/b$. General form can also represent vertical lines, where $b=0$.

The point-gradient form uses a known point and a gradient to determine a straight-line equation:

Substitute the known point and gradient first. Rearrange only if the question asks for another form.

All three forms describe the same geometric object, but each makes different information easier to see. Gradient-intercept form reveals the slope and vertical intercept. Point-gradient form is efficient when a point and gradient are known, while general form treats both variables symmetrically and can represent vertical lines. Converting between these forms reflects Descartes' central insight: geometric information can be encoded algebraically, and algebraic relations can be understood geometrically. Neither representation shows every insight immediately, so mathematical knowledge often develops by moving between them.

Parallel and perpendicular lines

Parallel lines are coplanar lines that never meet. For two distinct nonvertical lines,

Equal gradients give equal rates of change. If the lines are distinct, their intercepts must differ. Vertical lines are also parallel to one another, although their gradients are undefined.

Perpendicular lines intersect at a right angle. When both gradients are defined,

The second gradient is therefore the negative reciprocal of the first. A horizontal line and a vertical line are perpendicular, but their gradients can’t be checked with the product rule because the vertical gradient is undefined.

Gradients of inclines

For an incline, divide the vertical rise by the horizontal run—not by the distance along the sloping surface:

Metres divide by metres, so the gradient is dimensionless. For example, a gradient of $0.08$ means a rise of $0.08\ \text{m}$ for every $1\ \text{m}$ travelled horizontally.

Roads, bridges and access ramps may give the gradient as a percentage instead:

Don’t confuse percentage gradient with the angle of inclination.

This constant-rate idea also applies beyond physical inclines. In economics, a line may model an exchange rate or the relationship between price and quantity on a demand or supply graph. In experimental science, the gradient may show the change in a measured response per unit change in an independent variable. Its units matter because they show what the rate represents in that context.