91Ó°ÊÓ

The point-slope formula is a useful way to represent the equation of a line when you have one point on the line and the slope. The formula is expressed as \( y - y_1 = m(x - x_1) \). Here, \( (x_1, y_1) \) represents the coordinates of the specific point you know on the line, and \( m \) represents the line's slope, which is the rate of change between the x and y values.
To apply this formula to your exercise, we first identify the given point \( (2, -1) \) and the slope \( m = -3 \). By substituting these values into the formula, you create a base equation. This equation serves as your starting point to derive other forms of linear equations, providing a clear way to visualize the line if you have limited data, like just one point and a slope.

The slope-intercept form of a linear equation is a popular format because it allows you to easily identify both the slope and the y-intercept of a line. Written as \( y = mx + b \), \( m \) is the slope, while \( b \) represents the y-intercept, which is where the line crosses the y-axis.
In step 4 of the exercise solution, you converted the point-slope equation \( y + 1 = -3(x - 2) \) to its slope-intercept form by simplifying it to \( y = -3x + 5 \). This simplification process involves a few straightforward algebraic steps: distribute \( x \), and solve for \( y \). With the slope-intercept form, you can easily plot the y-intercept and use the slope to construct a line graph.

Plotting points is an essential skill for graphing linear equations. In this process, you start with known specific locations on a graph provided by coordinates, often called points. These points serve as reference markers to draw a straight line representing the linear relationship.
To graph the line using \( y = -3x + 5 \), start by identifying the y-intercept, which is \( (0, 5) \). This point is your start point. From there, use the slope of the line to find the next point; the slope \( -3 \) instructs you to move down 3 units and right 1 unit from the y-intercept to plot a second point. Connecting these points will give you the complete graph of your linear equation, ensuring accuracy in visual representation.

The slope of a line is a measure of its steepness and direction. It's calculated as the ratio of the vertical change to the horizontal change between two points on a line, often called "rise over run." The formula for the slope \( m \) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
In the given exercise, the slope is provided as -3. This negative value expresses that the line slopes downwards, meaning as you move right across the plane, the line decreases. The actual slope indicates how quickly the line drops; here, for every increase of 1 unit in x, y decreases by 3 units.

• A positive slope means the line rises as it moves from left to right.
• A negative slope means the line falls as it moves from left to right.
• If the slope is zero, the line is horizontal.
• An undefined slope corresponds to a vertical line.

Understanding the slope is crucial for predicting the behavior of lines in the coordinate plane and allows you to accurately graph and interpret linear data.