Problem 20 - Thinking Mathematically

Question

Calculate the slope of the line passing through the given points. If the slope is undefined, so state. Then indicate whether the line rises, falls, is horizontal, or is vertical. \((-2,3)\) and \((1,3)\)

Step-by-Step Solution

Answer

The slope of the line passing through points \((-2,3)\) and \((1,3)\) is \(0\). The line is horizontal.

Step 1: Identify the coordinates

Identify the coordinates of each given point. Point 1, labeled as \(A\), will have coordinates \((-2,3)\), represented as \((x_1, y_1)\). Point 2, labeled as \(B\), will have coordinates \((1,3)\), represented as \((x_2, y_2)\).

Step 2: Apply the slope formula

The slope (\(m\)) of the line is given by the formula \(m=\frac{y_2-y_1}{x_2-x_1}\). Plugging in the given coordinates, we have \(m=\frac{3-3}{1-(-2)}\).

Step 3: Simplify the result

Simplify the numerator and denominator. For the numerator, \(3-3 = 0\). For the denominator, \(1 - (-2) = 3\). This gives \(m=\frac{0}{3}\).

Step 4: Determine The Direction Of The Line

Since a slope of \(0\) means that the line is completely horizontal in direction, there is no rise or fall, and therefore this is a shape of a horizontal line.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study that combines algebra and geometry to solve problems involving points, lines, and shapes using a coordinate system. The coordinate plane consists of a horizontal axis, called the x-axis, and a vertical axis, called the y-axis. Every point on this plane has a unique set of coordinates, denoted as \(x, y\). These coordinates represent its position relative to the axes.

In our exercise, we have two points, \(A(-2, 3)\) and \(B(1, 3)\). These points are located on the coordinate plane, with the x-values indicating their horizontal positions and the y-values indicating their vertical positions. Understanding their placement is crucial for calculating slopes and understanding line orientations. Without coordinate geometry, it would be challenging to describe their relationship on a two-dimensional plane.

• Coordinates are essential in determining geometric relationships.
• Coordinates are expressed as ordered pairs \( (x, y) \).
• Each axis helps identify the position and movement along it.

Using this system helps identify characteristics like slopes, distances, and midpoints, making it an invaluable tool in mathematics.

Horizontal Line

A horizontal line is a straight line on the coordinate plane where all points have the same y-coordinate. It runs parallel to the x-axis. This property is what differentiates it from vertical or sloped lines. Such lines signify uniformity in vertical positioning, regardless of horizontal movement.

In the exercise, the given points A(-2, 3) and B(1, 3) share the same y-coordinate (3). This tells us that the line passing through these points is horizontal. When moving from point A to point B, only the x-coordinate changes, meaning there isn't any vertical movement—hence, the line is flat.

• All points on a horizontal line have identical y-values.
• Horizontal lines have a slope of zero since there is no vertical change.
• These lines are parallel to the x-axis.

A strong grasp of horizontal lines ensures accurate interpretations of linear graphs and enhances spatial reasoning.

Slope Formula

The slope formula is a fundamental concept in coordinate geometry that measures the steepness or inclination of a line. It is defined as the change in y-coordinates divided by the change in x-coordinates between two distinct points, formulated as \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula allows us to understand how much a line rises or falls as we move along it.

In our example, substituting the coordinates A(-2, 3) and B(1, 3) into the formula gives \( m = \frac{3 - 3}{1 - (-2)} = \frac{0}{3} \). This results in a slope of 0, confirming the line's horizontal nature. This indicates no rise, as the numerator is zero.

• The slope determines whether a line is rising, falling, or horizontal.
• Zero slope means the line is perfectly horizontal, as in our example.
• A positive slope means the line rises from left to right, while a negative slope falls.

The slope formula is essential for graphing linear equations, understanding motion, and solving various algebraic and geometric problems.