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When you talk about the slope of a line, you're referring to how steep the line is. The slope is essentially the measure of how much the line goes up or down as you move from one point to another. It's usually denoted by the letter "m". To find it between two points

• (x1, y1)
• (x2, y2)

you can use the formula: \[m = \frac{y2 - y1}{x2 - x1}\]This formula calculates the "rise" (the change in y) over the "run" (the change in x).

If the y-values of both points are the same, like in our example where both y-values are 4, then there's no change in height. Consequently, the slope is zero. This zero slope indicates a horizontal line, implying that no matter how far you move left or right, the height remains unchanged.

The slope-intercept form is a way to express the equation of a straight line using two elements: the slope and the intercept. It's written as:\[y = mx + c\]where:

• \(m\) is the slope
• \(c\) is the y-intercept

The y-intercept is where the line crosses the y-axis. With this form, you can easily identify both the slope and the initial starting point of the line on the graph.

From our example:

• Slope \(m\ = 0\)
• Y-intercept \(c = 4\)

Thus, the equation simplifies to \(y = 4\). This tells us that for any value of \(x\), \(y\) remains 4, confirming our line is indeed horizontal.

Coordinate Geometry, also known as Analytic Geometry, is a branch of geometry where you use algebra to describe and investigate geometrical shapes. It helps you understand spatial relationships using a coordinate system.

In a coordinate plane, every point is represented by an ordered pair

• \((x, y)\)

where \(x\) is the horizontal coordinate, and \(y\) is the vertical coordinate. By working within this framework, you can find distances between points, areas of shapes, and much more.

For our case, coordinate geometry helped identify that the points \((1,4)\) and \((3,4)\) not only lie on a horizontal line but also share the same \(y\)-coordinate. This allowed us to easily determine the equation of the line and understand its flat trajectory.