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An equation of a line provides a mathematical way to describe a straight line on a coordinate plane. One of the most common ways to express the equation of a line is in the slope-intercept form. This is written as \( y = mx + b \). Here, \( y \) and \( x \) represent the coordinates of points on the line, \( m \) is the slope of the line, and \( b \) is the y-intercept, which tells us where the line crosses the y-axis.

This form, \( y = mx + b \), is particularly useful because it allows us to quickly identify the slope and y-intercept from the equation itself. To find the equation of a line, we need to know two things:

• The slope \( m \)
• A point \((x, y)\) the line passes through

This information allows us to plug values into the formula and solve for the y-intercept \( b \), completing the equation of a line.

The slope of a line is a measure of its steepness. It is denoted by \( m \) in the slope-intercept form, \( y = mx + b \). The slope tells us how much the y-coordinate of a point on the line changes for a unit change in the x-coordinate. In mathematical terms, if you have two points \((x_1, y_1)\) and \((x_2, y_2)\), the slope \( m \) is calculated as:

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

A positive slope means the line ascends as it moves from left to right, while a negative slope means it descends. If the slope is zero, the line is horizontal, and if it's undefined, the line is vertical. In the provided solution, the slope \( m \) was given as 3, indicating a positively inclined line with a steep rise as you move along the x-axis.

Coordinates are used to pinpoint the exact location of a point on a two-dimensional plane, often referred to as the coordinate plane. A coordinate pair is represented as \( (x, y) \), where \( x \) is the horizontal distance from a reference line called the y-axis, and \( y \) is the vertical distance from a reference line called the x-axis.

In the context of the slope-intercept form, a known point's coordinates play an essential role. They allow us to substitute these values into the line equation to find any unknowns, such as the y-intercept. In the exercise, the point \((5, -\frac{1}{3})\) provided a specific example where these coordinates were crucial to calculating the y-intercept and ultimately writing the complete equation of the line.