Problem 13 - Precalculus: Mathematics for Calculus

Question

Find the slope of the line through \(P\) and \(Q .\) \(P(5,4), Q(0,4)\)

Step-by-Step Solution

Answer

The slope of the line through points \(P(5,4)\) and \(Q(0,4)\) is \(0\).

Step 1: Identify the Points

First, note the coordinates of the points. Here, point \(P\) has coordinates \((5,4)\) and point \(Q\) has coordinates \((0,4)\).

Step 2: Recall the Slope Formula

The formula to find the slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Step 3: Substitute the Coordinates

Substitute the coordinates of the points \(P\) and \(Q\) into the slope formula: \( m = \frac{4 - 4}{0 - 5} \).

Step 4: Simplify the Expression

Calculate the difference in the y-coordinates and the x-coordinates: \( m = \frac{0}{-5} = 0\).

Step 5: Interpret the Result

A slope of \(0\) indicates the line is horizontal, meaning the y-coordinates are equal for all points on the line.

Key Concepts

Coordinate geometry

Coordinate geometry is a branch of mathematics that describes geometric figures using a coordinate system. In this system, each point in a plane is defined by an ordered pair of numbers. These numbers are called coordinates and are usually represented as \((x, y)\). The horizontal axis is the x-axis, while the vertical axis is the y-axis.

Points are used to define lines and curves in coordinate geometry. For instance, the line described in the exercise goes through the points \(P(5,4)\) and \(Q(0,4)\), which are given their specific locations on the coordinate plane by their coordinates. Lines can be described and analyzed by looking at the relationships between these pairs of numbers.

Using coordinate geometry, we can easily find patterns, slopes, and even equations for lines, making it a powerful tool for solving mathematical problems and understanding the properties of different shapes.

Slope formula

The slope of a line in a coordinate plane is a measure of how steep the line is. It is denoted by the letter \(m\) and can be calculated using two points on the line. The formula to find the slope \(m\) is\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two points on the line.

The slope formula calculates the change in the y-values relative to the change in the x-values:

• \(y_2 - y_1\): the difference in the y-coordinates (rise)
• \(x_2 - x_1\): the difference in the x-coordinates (run)

This is often referenced as "rise over run."

By substituting specific point coordinates into this formula, such as \((5,4)\) and \((0,4)\) from our exercise, we see that the result is \(0\), indicating a horizontal line.

Horizontal line

A horizontal line in coordinate geometry is one that runs parallel to the x-axis. This means that all points on this line have the same y-coordinate. When using the slope formula, a horizontal line will always have a slope of \(0\) because there is no change in the y-values as you move along the line.

In the given exercise, the line passes through points \(P(5,4)\) and \(Q(0,4)\). Both these points share the same y-coordinate \(4\). When you calculate the slope using the formula, you end up dividing \(0\) by \(-5\), resulting in a slope of \(0\).

This property of a horizontal line is important because it signifies that no matter the distance along the x-axis, the height of the line remains constant. Students often encounter horizontal lines in graphs and must recognize that such lines indicate no vertical movement, highlighting the relationship between constant y-values and a slope of zero.