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Calculating the slope of a line is an important aspect when working with linear equations. The slope, often denoted by the letter \( m \), indicates the steepness and direction of a line. A line's slope is calculated using the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Let us break this down:

• \( x_1 \) and \( y_1 \) are the coordinates of the first point.
• \( x_2 \) and \( y_2 \) are the coordinates of the second point.
• The numerator \( y_2 - y_1 \) represents the change in the y-direction (vertical change).
• The denominator \( x_2 - x_1 \) represents the change in the x-direction (horizontal change).

When you plug the coordinates of your points into this formula, you determine how much the line moves up or down for each unit it moves left or right on the graph. In our case, with points \((-1, 5)\) and \((7, 5)\), the result is a slope of 0, indicating a perfectly horizontal line.

A horizontal line is a special type of line on the coordinate plane where all points have the same y-coordinate. This means it stretches endlessly from left to right but remains at a constant height vertically.
In our example with points \(P(-1, 5)\) and \(Q(7, 5)\), both points share the y-coordinate of 5, which confirms this line is horizontal.
Here's why horizontal lines are unique:

• Their slope \( m \) is always 0.
• The line has no vertical rise; it only runs horizontally.
• The equation for a horizontal line, using the slope-intercept form, simplifies to \(y = c\), where \(c\) is the y-coordinate shared by all points on the line.

Understanding this will help you quickly recognize horizontal lines simply by examining the y-coordinates of any two points on the line.

The y-intercept is a crucial parameter in the slope-intercept form of a linear equation and denoted as \( c \) in the equation \(y = mx + c\). This is the point where the line crosses the y-axis, providing insight into where the line is positioned vertically on the graph.

• For a horizontal line, the y-intercept is particularly easy to identify as it will be the same as the y-coordinate of the points on the line.
• In our example, both points \((-1, 5)\) and \((7, 5)\) have the same y-coordinate, therefore the y-intercept \( c \) is 5.

Thus, the equation of our line simplifies to \( y = 5 \), showcasing that the line is a constant provided by the y-coordinate, both when it meets the y-axis and along the entirety of the line itself.

A linear equation is a mathematical expression that models a straight line when graphed. The general form of these equations in two dimensions is known as the slope-intercept form, represented as \( y = mx + c \). Key aspects include:

• \( m \) represents the slope of the line, describing its angle.
• \( c \) is the y-intercept, indicating where the line crosses the y-axis.

For the problem of our given points, since the slope \( m \) is 0, the equation becomes \( y = 5 \). This indicates that no matter what value \( x \) takes on, \( y \) remains 5. The line does not rise or fall but stays flat across the y-axis at 5.
Linear equations like this emphasize consistency and reliability in their predictability since any \( x \) input will always yield the same \( y \) output on a horizontal line. Understanding how a linear equation functions is pivotal for interpreting data and predicting outcomes.