91Ó°ÊÓ

The Slope-Intercept Form is a widely used equation format for straight lines and is written as \(y = mx + b\). This formula is extremely helpful because it immediately gives us two key pieces of information:

• Slope (\(m\)): This tells us the angle or steepness of the line
• Y-intercept (\(b\)): This is where the line crosses the y-axis

The Slope-Intercept Form makes it easy to understand and manipulate equations of lines, especially when determining relationships between different lines. It is essential in identifying key attributes like slopes and intercepts just by looking at the equation.

The slope of a line is a measure of its steepness and is often represented by the letter \(m\). The slope indicates the degree to which the line inclines or declines.

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

In our example, the given line is written as \(y = -\frac{1}{3}x + \frac{4}{3}\). Here, the slope \(m\) is \(-\frac{1}{3}\), which means the line falls gently as you move along.

The y-intercept of a line is the point where the line crosses the y-axis. It is embodied by the term \(b\) in the slope-intercept equation \(y = mx + b\). This is a very useful piece of information because it defines a starting point for our line, or where it "begins" on the graph.In the example, the y-intercept of the original line \(y = -\frac{1}{3}x + \frac{4}{3}\) is \(\frac{4}{3}\). This tells us that the line crosses the y-axis at a vertical position of \(\frac{4}{3}\). Understanding the y-intercept helps in constructing and visualizing the complete line when combined with the slope.

The equation of a line succinctly describes its direction and position on the coordinate plane. There are several ways to represent this, but the Slope-Intercept Form \(y = mx + b\) is often the most convenient.Having both the slope and y-intercept enables us to write this equation simply. For instance, to find a line parallel to another, we adhere to the rule that parallel lines share the same slope. So if we know a parallel line's slope, we need only one point that the new line passes through to find \(b\), our y-intercept.In our problem, after calculating the correct y-intercept using the point \(P(1, -1)\), we arrive at the final equation \(y = -\frac{1}{3}x - \frac{2}{3}\). This line runs parallel to our original line and passes through a specific point, demonstrating how equations can be manipulated to achieve desired line attributes.