Problem 49 - Calculus with Concepts in Calculus

Question

Let \(l\) be the line that contains two given points, \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\), with \(x_{1} \neq x_{2} .\) Show that an equation of \(l\) is $$ y-y_{1}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\left(x-x_{1}\right) $$ (This equation is called a two-point equation of \(l\).)

Step-by-Step Solution

Answer

The equation of the line through points \\( (x_1, y_1) \\\) and \\( (x_2, y_2) \\\) is \\( y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) \\\).

Step 1: Understanding the Two Points Given

We have two points on the line, \( (x_1, y_1) \) and \( (x_2, y_2) \), with the condition \( x_1 eq x_2 \). The purpose is to derive the equation of the line passing through these points.

Step 2: Calculate the Slope of the Line

The slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This slope represents the rate of change of \( y \) with respect to \( x \).

Step 3: Use Point-Slope Form of the Line Equation

The point-slope form of the equation of a line is given by \[ y - y_1 = m(x - x_1) \] where \( m \) is the slope, and \( (x_1, y_1) \) represents a point on the line. Since we've already calculated the slope in Step 2, we substitute it into this formula.

Step 4: Substitute the Slope into Point-Slope Form

Substitute the expression for the slope from Step 2: \[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) \] This equation directly represents the line in two-point form, since it incorporates both points \( (x_1, y_1) \) and \( (x_2, y_2) \).

Key Concepts

Line Equation Derivation

Deriving a line equation involves finding a mathematical expression that describes all the points on a line. This expression is a function of two variables, usually \( x \) and \( y \), which represents a line's coordinates in a flat plane. For a straight line, one common form of this equation is the slope-intercept form, \( y = mx + b \), where \( m \) denotes the slope and \( b \) is the y-intercept. However, when dealing with two given points, as in the original exercise, we often use what's called the two-point form.

The two-point form equation is derived by using these specific points to find the line equation. By understanding how the equation is structured, it's easier to see how it characterizes a straight line in terms of its slope and a chosen point. Once you determine the slope between the two points, substituting it into the point-slope form yields a precise line equation that fully agrees with the given points on the line.

Slope

The slope of a line is a measure of its steepness, often represented by the letter \( m \). To find the slope of a line given two distinct points \( (x_1, y_1) \) and \( (x_2, y_2) \), we use the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Think of the slope as the rise over run, which defines how much the line rises vertically for a given horizontal movement between two points.

• If the slope is positive, the line ascends from left to right.
• If it's negative, the line descends from left to right.
• A slope of zero implies a horizontal line, while an undefined (division by zero) slope indicates a vertical line.

The calculation of the slope is crucial as it forms the base for deriving the line's equation. This tells us how the y-coordinate changes with respect to the x-coordinate, providing a crucial link between the geometry and algebra of lines.

Point-Slope Form

The point-slope form of a line equation is a versatile way to express a line's equation using its slope and a specific point on the line. The general form of this equation is:

\[ y - y_1 = m(x - x_1) \]

Here, \( m \) represents the slope, and \( (x_1, y_1) \) is a known point on the line.

• This form is particularly useful when you know one point on the line and its slope, as it provides a straightforward method to write the equation quickly.
• To transition to the familiar slope-intercept form \( (y = mx + b) \), simply solve for \( y \).

The point-slope form fits perfectly within our context, where we derived the line's slope from two known points. Plugging this slope into the point-slope formula gives us the full equation for the line.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to describe geometric principles. In coordinate geometry, points are defined in terms of their coordinates, \( (x, y) \), on a plane, often called the Cartesian plane.

This domain extensively employs coordinates and equations to resolve geometric problems and describe shapes, like lines and curves, algebraically. Lines, for example, are easily described with equations like the ones we've discussed, using fundamental concepts such as slope and distance between points.

• It provides tools to compute distances, slopes, and even to derive equations of lines and curves.
• By converting diagrams and shapes into numbers and algebraic expressions, coordinate geometry makes it possible to apply algebraic methods to geometric problems.

Understanding coordinate geometry is essential, as it bridges the visual understanding of geometry with the precision of algebra, offering powerful methods to investigate and solve complex problems.