91Ó°ÊÓ

A linear equation can be written in several formats, but the slope-intercept form is one of the most common and useful. It looks like this: \( y = mx + c \). In this equation, \( m \) is the slope of the line, and \( c \) is the y-intercept (the point where the line crosses the y-axis). Knowing this form allows us to quickly and easily identify the key characteristics of the line.
For the given exercise, the equation is written as: \[ y = \frac{9}{5}x - 4 \] Before even starting with the exercise, it's crucial to understand that we can directly compare this equation with the slope-intercept form \( y = mx + c \) to extract important information, like the slope and the y-intercept, without the need for further transformation.

The slope of a line measures how steep the line is. It indicates the rate at which \( y \) changes for a unit change in \( x \). Mathematically, the slope is represented by \( m \) in the slope-intercept form \( y = mx + c \).
The slope tells us whether the line ascends (if \( m \) is positive), descends (if \( m \) is negative), or is horizontal (if \( m \) is 0). For this exercise, comparing the given equation to the slope-intercept form, we found that \( m = \frac{9}{5} \).
This means for every 5 units that we move horizontally (along the x-axis), the line rises 9 units vertically (along the y-axis). Hence, the line is ascending. Understanding the slope helps in graphing the line and predicting the behavior of variables in relation to each other.

The y-intercept of a line is the value where the line crosses the y-axis. In simple terms, it is the point on the y-axis where \( x \) is zero. In the slope-intercept form \( y = mx + c \), the y-intercept is represented by \( c \).
For our exercise, we compared the given equation to \( y = mx + c \) and determined that \( c = -4 \). This means the line crosses the y-axis at \( -4 \).
Understanding the y-intercept is essential because it gives us a starting point for plotting the line on a graph. It shows where the line will start on the y-axis, and together with the slope, it allows us to sketch the line accurately.