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The slope of a line is a measure of its steepness and direction. Calculating the slope is a fundamental skill in understanding linear equations. The slope is commonly denoted by the letter \( m \). It is found using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This formula calculates the change in the y-values divided by the change in the x-values between two points on a line. In simpler terms, it tells us how much the y-coordinate (vertical change) increases or decreases per unit change in the x-coordinate (horizontal change).

• If \( m > 0 \), the line slopes upwards.
• If \( m < 0 \), the line slopes downwards.
• If \( m = 0 \), the line is horizontal.

In the given exercise, using the points (-2,8) and (4,6), substitute them into the formula:

• Calculate the change in \( y \): \( 6 - 8 = -2 \)
• Calculate the change in \( x \): \( 4 - (-2) = 6 \)
• Slope \( m = \frac{-2}{6} = -\frac{1}{3} \)

This slope of \( -\frac{1}{3} \) indicates a line that descends to the right.

Once we have calculated the slope, the next step is to understand and use the point-slope form of a line. The point-slope form of a linear equation is useful because it allows us to create an equation when we know:

• A point on the line, \((x_1, y_1)\)
• The slope \(m\) of the line

The point-slope form is written as: \[ y - y_1 = m(x - x_1) \] This formula provides a way to express the equation of a line that passes through a specific point with a given slope. Using our points (-2, 8) and the slope \( -\frac{1}{3} \) from the previous section, we substitute these values into the point-slope formula: \[ y - 8 = -\frac{1}{3}(x + 2) \]. This equation reflects that the line passes through the point (-2,8) with a slope of \(-\frac{1}{3}\). This form is particularly helpful because once the basic information is plugged in, you have an expression for the line's equation that you can easily convert to other forms.

The y-intercept is the point where the line crosses the y-axis. In the linear equation \( y = mx + b \), \( b \) is known as the y-intercept. It tells us the value of \( y \) when \( x = 0 \). To find the y-intercept, we often rearrange our line's equation into the slope-intercept form if it isn’t already. After using point-slope form, we simplify to find the classic \( y = mx + b \). For the equation derived in this task, \( y - 8 = -\frac{1}{3}(x + 2) \), simplifying gets us: \[ y = -\frac{1}{3}x - \frac{2}{3} + 8 \] Combine the constant term: \( y = -\frac{1}{3}x + \frac{24}{3} - \frac{2}{3} \) which simplifies further to: \( y = -\frac{1}{3}x + \frac{22}{3} \). Thus, the y-intercept \( b \) is \( \frac{22}{3} \). This means when \( x = 0 \), the value of \( y \) is \( \frac{22}{3} \). Knowing the y-intercept helps in graphing the line as it gives a starting point. It is also crucial in comparing lines and understanding how they are positioned relative to one another on a graph.