91Ó°ÊÓ

Understanding the concept of the slope is essential when working with linear equations. The slope of a line, often denoted as m, represents how steep the line is. To calculate the slope between two points \((x_1, y_1) \) and \((x_2, y_2) \), use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here’s a step-by-step breakdown:

• Subtract the y-coordinate of the first point from the y-coordinate of the second point. This gives you the rise or vertical change.
• Subtract the x-coordinate of the first point from the x-coordinate of the second point. This gives you the run or horizontal change.
• Finally, divide the rise by the run.

In our example, with points \((-5, -3) \) and \((4, -6)\), the calculations are: \[ m = \frac{-6 - (-3)}{4 - (-5)} = \frac{-6 + 3}{4 + 5} = \frac{-3}{9} = -\frac{1}{3} \] So, the slope of the line is \(- \frac{1}{3}\).

After determining the slope, we use the point-slope form of a line's equation to find the equation of the line. The point-slope form is given by: \[ y - y_1 = m(x - x_1) \] Here, \((x_1, y_1)\) is any point on the line, and m is the slope. In our problem, we have the point \((-5, -3)\) and the slope \(- \frac{1}{3}\). To apply the formula: \[ y - (-3) = -\frac{1}{3}(x - (-5)) \] Simplifying this, we get: \[ y + 3 = -\frac{1}{3}(x + 5) \] This is the equation of the line in point-slope form. This form is particularly useful for finding the equation when you already know a point on the line and the slope.

Converting the equation to slope-intercept form makes it easy to understand the line's behavior and graph it. The slope-intercept form of a line's equation is: \[ y = mx + b \] where m is the slope and b is the y-intercept. Using the point-slope form from the previous step: \[ y + 3 = -\frac{1}{3}(x + 5) \] Distribute the slope on the right side: \[ y + 3 = -\frac{1}{3}x - \frac{5}{3} \] Next, isolate y by subtracting 3 from both sides: \[ y = -\frac{1}{3}x - \frac{5}{3} - 3 \] Convert the constants to a common denominator and combine them: \[ y = -\frac{1}{3}x - \frac{5}{3} - \frac{9}{3} \] Simplify: \[ y = -\frac{1}{3}x - \frac{14}{3} \] The equation in slope-intercept form is: \[ y = -\frac{1}{3}x - \frac{14}{3} \]. Now, you can easily understand that the line has a slope of \(- \frac{1}{3}\) and crosses the y-axis at \(- \frac{14}{3}\).