Linear equations

The study of analytic geometry begins with the fundamental concept of the straight line, which serves as the foundation for understanding more complex geometric relationships. A straight line in the Cartesian coordinate system can be represented through several equivalent mathematical forms, each offering unique advantages for different applications.

Definitions

Cartesian coordinate system

The Cartesian coordinate system is a two-dimensional plane defined by perpendicular axes ($x$ and $y$), used to locate points and express geometric figures algebraically.

Slope

The slope of a line measures its steepness, defined as the ratio of vertical change to horizontal change between two points on the line.

Intercept

An intercept is the point where a line crosses an axis.

Perpendicular

Two lines are perpendicular when they intersect at a right angle ($90°$), which occurs when the product of their slopes equals $–1$.

Distance

The distance between two points in the plane is the length of the straight line segment connecting them, calculated using the Pythagorean theorem.

Slope-intercept form

The slope-intercept form, also known as the standard form, provides an intuitive understanding of a line’s behavior:

$$y=mx+b$$

In this equation, $m$ represents the slope of the line, which measures the rate of change of $y$ with respect to $x$. The parameter $b is the $y$-intercept, indicating where the line crosses the $y$-axis.

The slope $m$ can be interpreted as the vertical change (rise) divided by the horizontal change (run) between any two points on the line. A positive slope indicates the line rises from left to right, while a negative slope indicates the line falls from left to right. A slope of zero represents a horizontal line, while an undefined slope corresponds to a vertical line.

A line in slope-intercept form with slope $m$ and $y$-intercept $b$ is shown in the figure below:

@@@ Figure [linear-1]: Straight line in slope-intercept form @@@

Point-slope form

When a point on the line and the slope are known, the point-slope form provides a direct method for writing the line’s equation:

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

@@@ Figure [linear-2]: Point-slope form @@@

As shown in the figure above, $(x_1,y_1)$ represents a known point on the line, and $m$ is the slope. This form is particularly useful in applications where specific points and rates of change are given.

Slope calculation from two points

Consider two distinct points $(x_1,y_1)$ and $(x_2,y_2)$ on a line as shown below:

@@@ Figure [linear-3]: Two points on a line @@@

The slope can be calculated using:

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

This formula represents the fundamental definition of slope as the ratio of vertical change to horizontal change. It is important to note that this formula is undefined when $x_2=x_1$, which corresponds to a vertical line.

General form of a line

Another representation of a straight line is given by the general form equation:

$$Ax+By+C=0$$

Where $A$, $B$, and $C$ are constants, and not both $A$ and $B$ are zero. This form is particularly useful because it can represent any line in the plane, including vertical lines where the slope is undefined.

The coefficients in the general form have geometric significance. The coefficient $A$ represents the change in the $x$-direction, while $B$ represents the change in the $y$-direction. The constant term $C$ determines the line’s position relative to the origin.

Angle between two intersecting lines

Consider two intersecting lines with slopes $m_1$ and $m_2$, and with an angle $\alpha$ between them:

@@@ Figure [linear-4]: Two intersecting lines @@@

The angle $\alpha$ between them can be determined using the arctangent function:

$$\alpha =\arctan \left[\frac{m_2-m_1}{1+m_2\cdot m_1}\right]$$

This formula derives from the tangent addition formula and provides the acute angle between the two lines. The expression is valid when $1+m_1{m}_2\ne 0$. When $1+m_1{m}_2=0$, the lines are perpendicular.

Perpendicular lines

Two lines are perpendicular when the product of their slopes equals negative one:

$$m_1=-\frac{1}{m_2}$$

This relationship, known as the negative reciprocal condition, ensures that the angle between the lines is exactly 90 degrees.

Distance between two points

@@@ Figure [linear-5]: Two points and the distance between them @@@

The distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ in the Cartesian plane is given by the Pythagorean theorem:

$$d=\sqrt{(y_2-y_1{)}^2+(x_2-x_1{)}^2}$$

This formula represents the length of the straight line segment connecting the two points, and may be generalized to any number of spatial dimensions.