The Equation of a Line

Part 11, Chapter 11: Lines, Rays, and Segments

Learning objectives

Write the equation of a line in slope-intercept form and point-slope form
Find the slope of a line through two given points
Identify horizontal and vertical lines and their equations
Convert between different forms of line equations

The parametric form is elegant; the ordinary form $y = mx + b$ is what you grew up with. Both describe the same lines, but the ordinary form trades the parameter $t$ for a direct relation between $x$ and $y$ . That makes it perfect for graphing, for spotting whether two lines are parallel or perpendicular at a glance, and for the entire enterprise of school algebra. This section is the bridge between the two views: parametric and ordinary.

Slope, the rate of change

The slope of the line through $(x_1, y_1)$ and $(x_2, y_2)$ (assuming $x_1 \neq x_2$ ) is

$m = \frac{y_2 - y_1}{x_2 - x_1}.$

It measures how much $y$ changes for each unit change in $x$ . A slope of $2$ means: walk $1$ unit right, climb $2$ units. A slope of $-1/3$ means: walk $3$ units right, drop $1$ unit. Slope is the geometric soul of the line, and (in calculus) it generalises to the derivative.

Slope-intercept form

The most common equation for a line:

$y = m x + b.$

Here $m$ is the slope and $b$ is the $y$ -intercept, the value of $y$ where the line crosses the $y$ -axis (i.e., when $x = 0$ ). Two numbers determine a line completely.

Where this shows up

Statistics: Linear regression fits the line $y = mx + b$ to a cloud of data points; the slope $m$ is the most-asked-for number in elementary stats ("how much does Y change per unit of X?").

Economics: A demand curve $Q = a - bP$ is a line in slope-intercept form; the slope $-b$ measures price sensitivity (price elasticity at a point), one of the most central quantities in microeconomics.

Physics: Hooke's law $F = -kx$ is a line through the origin with slope $-k$ ; the spring constant is just the negative slope, and finding $k$ from data is the simplest experiment a physics student does.

Adjust the slope and intercept sliders. The equation display updates live as $y = mx + b$ . Each pair $(m, b)$ gives a unique line.

Point-slope form

When you know the slope $m$ and any one point $(x_1, y_1)$ on the line, the point-slope form writes the equation directly:

$y - y_1 = m (x - x_1).$

This is just the slope formula $m = (y - y_1)/(x - x_1)$ rearranged. It is the workhorse for finding line equations: compute slope, plug in a point, simplify if you want.

Special cases: horizontal and vertical lines

A horizontal line has slope $m = 0$ ; its equation is $y = b$ (constant).

A vertical line has undefined slope (division by zero in $m$ ); its equation is $x = a$ . Vertical lines cannot be written as $y = mx + b$ .

Parallel and perpendicular: slope tests

Using slopes, two non-vertical lines are:

Parallel ⇔ their slopes are equal: $m_1 = m_2$ .

Perpendicular ⇔ their slopes are negative reciprocals: $m_1 m_2 = -1$ , or $m_2 = -1/m_1$ .

So $y = 3x + 2$ and $y = 3x - 5$ are parallel (same slope $3$ , different intercept); $y = 2x + 1$ and $y = -\tfrac{1}{2}x + 7$ are perpendicular ( $2 \cdot (-1/2) = -1$ ).

From two points to an equation, the standard recipe

Through $(1, 3)$ and $(4, 9)$ : slope $m = (9 - 3)/(4 - 1) = 2$ . Use point-slope with $(1, 3)$ : $y - 3 = 2(x - 1)$ . Simplify: $y = 2x + 1$ . Done. The same line can be written as $y - 9 = 2(x - 4)$ using the other point, check by simplifying that you get $y = 2x + 1$ either way.

Try it

Before adjusting: with $m = 2$ and $b = 0$ , what is the equation of the line, and which point does it pass through? Set the sliders and verify it is $y = 2x$ through the origin.

Predict first: with $m = -1$ and $b = 3$ , where does the line cross the $y$ -axis, and does it rise or fall? Set the sliders and verify it is $y = -x + 3$ .

What is the slope of a line perpendicular to $y = \tfrac{2}{5}x + 1$ ? Answer: $-\tfrac{5}{2}$ .

A trap to watch for

"Parallel lines have the same equation." No, parallel lines have the same slope but generally different intercepts. The lines $y = 2x + 1$ and $y = 2x + 5$ are parallel but distinct (the second one is shifted up by $4$ ). If two distinct lines have the same equation, they are the same line, which is more than parallel. A second common trap: forgetting that vertical lines have no slope and cannot fit $y = mx + b$ . Always check whether the two points share an $x$ -coordinate before computing slope; if they do, you have a vertical line $x = a$ .

What you now know

You can move freely between the slope-intercept form $y = mx + b$ , the point-slope form $y - y_1 = m(x - x_1)$ , and the parametric form $P(t) = A + t(B - A)$ ; you can identify parallel and perpendicular lines from their slopes; and you can write the equation of a line through any two given points. With this section you have a complete algebraic toolkit for straight lines in the plane, the foundation for everything that follows in trigonometry, conics, and calculus.

Quick check

Mark section complete →

References

Lang, S. (1971). Basic Mathematics. Springer. Chapter 10, §4, the ordinary equation of a line and slope-intercept form.

Stewart, J. (2015). Calculus, 8th ed. Cengage. Appendix B covers lines in coordinates as a calculus prerequisite.

Apostol, T. M. (1967). Calculus, Volume 1, 2nd ed. Wiley. §1.6 develops slope-intercept lines and connects them to the derivative.

Coxeter, H. S. M. (1969). Introduction to Geometry, 2nd ed. Wiley. §13 develops line equations in the broader affine setting.

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