91Ó°ÊÓ

When we talk about an equation of a line in mathematics, we are often referring to a specific way to express this line using a formula. The most common form is the slope-intercept form, represented as \( y = mx + b \). This is a straightforward way to describe a line on the Cartesian plane. Here, "\( y \)" and "\( x \)" are variables representing the coordinates of any point on the line. Meanwhile, "\( m \)" is the slope of the line, and "\( b \)" is the \( y \)-intercept. This form makes it easy to quickly identify both the slope and \( y \)-intercept of the line.

When given a specific slope and \( y \)-intercept, you can plug these values directly into the equation to find the exact equation of that line. This clarity and simplicity are what make the slope-intercept form so popular in algebra.

The slope of a line is a measure of how steep the line is. It indicates the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line.

Here are some key points about the slope:

• If the slope is positive, the line rises as it moves from left to right.
• If the slope is negative, like in our example where \( m = -7 \), the line falls as it moves from left to right.
• A slope of zero indicates a horizontal line, where there is no rise or fall.

A steeper slope means a steeper line, and in our exercise, a slope of \(-7\) indicates a strong downward slope. The value \

The \( y \)-intercept of a line is the point where the line crosses the \( y \)-axis. It is represented by the \( b \) in the equation \( y = mx + b \).

Understanding this concept is crucial because:

• The \( y \)-intercept allows you to easily determine where the line will start on the graph.
• It is the value of \( y \) when \( x = 0 \).
• In our example, with a \( y \)-intercept of 2, the line crosses the \( y \)-axis at the point \((0, 2)\).

This point is often used to help graph the line, as starting from the \( y \)-intercept and following the slope gives a complete picture of the line’s direction and position on the graph.