Problem 37 - Elementary Algebra

Question

Sketch the line determined by each pair of points and decide whether the slope of the line is positive, negative, or zero. $$(-2,4),(6,4)$$

Step-by-Step Solution

Answer

The slope of the line is zero.

Step 1: Identify the coordinates

We have two points given: Point 1: \((-2, 4)\)Point 2: \((6, 4)\)

Step 2: Recognize horizontal change

Since both points have the same y-coordinate, \(y = 4\), there is no vertical change. This indicates a horizontal line passing through both points.

Step 3: Calculate the slope

The formula for calculating the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\(m = \frac{y_2 - y_1}{x_2 - x_1}\).Substituting our points: \ \(m = \frac{4 - 4}{6 - (-2)} = \frac{0}{8} = 0\).

Step 4: Drawing Conclusion

Since the slope \(m = 0\), it indicates that the line formed is horizontal.

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of geometry where we use coordinates to represent and analyze geometric shapes and figures in a plane. In the context of graphing linear equations, coordinate geometry allows us to work with points, lines, and other shapes by using a grid called the 'coordinate plane'. This grid is defined by two perpendicular axes: the x-axis (horizontal) and y-axis (vertical). Each point on the plane can be represented as an ordered pair

• The first number in the pair indicates the x-coordinate, which determines the point's position on the x-axis.
• The second number is the y-coordinate, indicating the point's position on the y-axis.

In our exercise, the points o (-2, 4) o (6, 4) lie on the coordinate plane, specifically on the same horizontal line because they share the same y-coordinate. Understanding coordinate geometry is key to easily plotting these points and visualizing the relationship between them.

Slope of a Line

The slope of a line measures its steepness. It tells us how much the line rises or falls as we move along it from left to right. To find the slope: Calculate the difference in the y-coordinates ("rise") and the x-coordinates ("run") between two points. Use the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\] Here, \(m\) represents the slope.In our original exercise, substituting the given points: * \(y_2 = 4\), \(y_1 = 4\), thus \(y_2 - y_1 = 0\), meaning no vertical change.* \(x_2 = 6\), \(x_1 = -2\), thus \(x_2 - x_1 = 8\), which represents a horizontal change.The formula yields a slope of \(0\), signifying a perfectly horizontal line. Understanding slopes helps identify how lines look on the graph and their direction.

Horizontal Line

A horizontal line runs left to right across the coordinate plane. It is characterized by having a constant y-coordinate, meaning there is no vertical change as you move along the line. In simpler terms: - If you pick any two points on a horizontal line, the y-values will be identical.- Because there is no vertical change, the slope is always \(0\).In the exercise, the points * \((-2, 4)\)* \((6, 4)\) illustrate this concept effectively. Both points share the same y-value, forming a horizontal line at \(y = 4\). Grasping the properties of horizontal lines is vital in graphing because it helps quickly determine the equation of the line and understand its slope.