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The slope of a line is a measure of its steepness and direction. It's represented by the letter \( m \) in mathematical equations.
You can calculate the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) using the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This formula gives the rate at which \( y \) changes with respect to \( x \). A positive slope means the line rises as it moves from left to right. A negative slope means it falls. A slope of zero indicates a horizontal line, and an undefined slope indicates a vertical line.
In our original exercise, the slope between the points \((2, -1)\) and \((0, 1)\) is calculated as \( m = -1 \). This tells us the line is declining as it moves from the first point to the second.

The slope-intercept form is a way of writing the equation of a line, making the slope and the y-intercept easy to identify. It is expressed as:

• \( y = mx + b \)

Here, \( m \) represents the slope, and \( b \) is the y-intercept, the point where the line crosses the y-axis. This format makes it simple to graph a line and understand its behavior at a glance.
To convert the equation of a line from point-slope form to slope-intercept form, distribute and simplify as necessary.
In our exercise, after simplifying \( y - (-1) = -1(x - 2) \), we find \( y = -x + 1 \). This equation is in slope-intercept form, where the slope \( m = -1 \) and the y-intercept \( b = 1 \).

Coordinate geometry, also known as analytic geometry, connects algebra and geometry through graphs and equations. It allows us to use algebraic methods to solve geometric problems by using coordinates on a graph.
Key concepts in coordinate geometry include:

• Points, defined by coordinates \((x, y)\)
• Lines, which can be represented by linear equations
• Distance and midpoint formulas for measuring and locating points

In the context of our exercise, coordinate geometry allows us to find the equation of a line passing through two specific points. By determining the slope and expressing the relationship between \( x \) and \( y \) in equation form, we effectively bridge the gap between numerical data and spatial representation.