91Ó°ÊÓ

The slope of a line is a measure of its steepness. It tells us how much the line rises or falls as it moves from left to right. In mathematical terms, the slope (\( m \)) is the change in the \( y \)-values divided by the change in the \( x \)-values between two points on the line.

For example, if the line goes up 3 units for every 1 unit it goes across, the slope is 3. This number can be positive, negative, zero, or undefined, depending on the angle of the line:

• A positive slope (like 3) means the line rises as it goes from left to right.
• A negative slope means the line falls as it moves to the right.
• A zero slope indicates a horizontal line.
• An undefined slope corresponds to a vertical line, where \( x \) does not change.

Understanding the slope is key to graphing a line accurately. It helps in predicting how the line will look on a graph.

The \( y \)-intercept is where the line crosses the \( y \)-axis on a graph. This point is important because it gives us a specific location where \( x \) is zero. When a line is expressed in slope-intercept form \( y = mx + b \), the \( y \)-intercept is represented by \( b \).

In the given exercise, we found the \( y \)-intercept by substituting the given point and the slope into the equation. From the point \((-2,1)\) and slope \(3\), we solved the equation \(1 = 3(-2) + b\) for \( b \). This means our line crosses the \( y \)-axis at \( 7 \). This intercept tells us where to start the line before using the slope to find other points on the line. It's a foundational step in drawing or understanding any linear equation.

The equation of a line in slope-intercept form is expressed as \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept. This format is popular because it clearly displays both important aspects: the slope and where the line intersects the \( y \)-axis.

To find the equation of a line when the slope and a point on the line are known, like in the exercise, you substitute these values directly into this form:

• Insert the slope (\( m \)).
• Use the given point to solve for the \( y \)-intercept (\( b \)).
• Combine them into the final equation.

For the given example with slope \( 3 \) and point \((-2, 1)\), the final equation of the line will be \( y = 3x + 7 \). Knowing and understanding this equation allows you to graph the line or use it to predict values within the linear relationship.