91Ó°ÊÓ

Slope is a fundamental concept in the study of linear equations. It describes how steep a line is and can be thought of as the "rise" over the "run". This involves comparing how much a line moves up or down (the rise) for each step it moves sideways (the run). The formula to find the slope is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]where

• \( (x_1, y_1) \)
• and
• \( (x_2, y_2) \) are the coordinates of two given points on the line.

For example, if you have the points \((-2, 4)\) and \((0, -6)\), you can plug these into the formula: \[ m = \frac{-6 - 4}{0 - (-2)} = \frac{-10}{2} = -5 \]Hence, the slope is \(-5\). This negative value indicates the line decreases (or goes downward) from left to right. When dealing with slopes, remember:

• A positive slope means the line is rising.

• A negative slope means the line is falling.

• A slope of zero means the line is horizontal.

• An undefined slope usually represents a vertical line.

Once you've figured out the slope of a line, the next step is to write the equation of the line using the point-slope form. This is extremely useful because it allows you to write the equation with just one point on the line and the slope. The point-slope form is expressed as:\[ y - y_1 = m(x - x_1) \]Here,

• \( m \) is the slope you calculated.
• \((x_1, y_1)\) is any point on the line, often one of the points you used to calculate the slope.

For example, using the slope \(-5\) and choosing the point \((-2, 4)\), the equation would be: \[ y - 4 = -5(x + 2) \]This step is powerful because it connects the abstract concept of slope with tangible points on a graph. You will find that using the point-slope form is quite flexible, especially when you need to switch points or change your understanding of the line.

• It is particularly handy in situations where you do not want to simplify to slope-intercept form immediately.
• Useful when working directly with graphs or need to modify linear equations quickly.

The final step involves simplifying your point-slope equation into the more familiar slope-intercept form, which appears as \( y = mx + b \). This form makes it straightforward to see the slope \( m \) and the y-intercept \( b \), which is where the line crosses the y-axis.For the line between points \((-2, 4)\) and \((0,-6)\), after determining the point-slope equation \( y - 4 = -5(x + 2) \), you distribute and simplify:

• First, distribute \(-5\) across \(x + 2\): \[ y - 4 = -5x - 10 \]
• Next, add \(4\) to both sides:\[ y = -5x - 10 + 4 \]
• Which simplifies into \[ y = -5x - 6 \]

This is your line segment equation in slope-intercept form. The slope \(-5\) is consistent across the line, confirming it's a straight path. The intercept \(-6\) indicates this line would cross the y-axis at \(-6\) if extended indefinitely. Keep these steps in mind for drawing or interpreting linear graphs:

• The slope-intercept form translates well into real life problems, making linear equations easier to visualize and graph.