Problem 5 - Algebra and Trigonometry

Question

Find the slope of the line through \(P\) and \(Q .\) \(P(2,4), Q(4,3)\)

Step-by-Step Solution

Answer

The slope is -\frac{1}{2}.

Step 1: Understanding the Slope Formula

The slope of a line through two points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) is given by the formula \m = \frac{y_2-y_1}{x_2-x_1}\.

Step 2: Substitute the Given Points

Using the points \(P(2,4)\) and \(Q(4,3)\), substitute into the formula: \(m = \frac{3-4}{4-2}\).

Step 3: Simplify the Expression

Calculate the difference in the y-values: \(3 - 4 = -1\). Calculate the difference in the x-values: \(4 - 2 = 2\). The expression becomes: \(m = \frac{-1}{2}\).

Step 4: Conclude the Calculation

The slope of the line through points \(P(2,4)\) and \(Q(4,3)\) is \m = -\frac{1}{2}\.

Key Concepts

Points and Coordinates

In mathematics, understanding points and coordinates is foundational. Points are specific locations on a graph, represented by pairs of numbers known as coordinates. Each point has two coordinates: an x-coordinate and a y-coordinate. These tell us how far the point is along the horizontal (x-axis) and vertical (y-axis) directions from the origin, which is the point (0,0) where the two axes intersect. For example, in the point \((2,4)\):

• The x-coordinate is 2, meaning it is 2 units away from the y-axis.
• The y-coordinate is 4, indicating it is 4 units above the x-axis.

Points like these help us plot lines and curves by providing precise locations on our graph. Knowing how to find and use these coordinates is crucial for solving many geometry and algebra problems, such as determining lines' slopes.

Slope Formula

The concept of the slope of a line is essential in algebra and coordinate geometry. The slope tells us how slanted a line is, or its 'steepness.' It is calculated using a simple formula, which involves two points on the line, \((x_1, y_1)\) and \((x_2, y_2)\). The slope formula is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula measures the vertical change (\(y_2 - y_1\), also known as the "rise") over the horizontal change (\(x_2 - x_1\), known as the "run").

Using this formula, we can determine how much a line moves up or down as it moves left to right. A positive slope means the line rises as it moves to the right, while a negative slope indicates it falls. If the difference in the y-coordinates (rise) is positive and the difference in x-coordinates (run) is positive, the line goes upwards; if the rise is negative and the run is positive, the line goes downwards.

Linear Equations

Linear equations represent straight lines on a graph, and understanding their structure is key to working with them. A standard form of a linear equation is \(y = mx + b\), where:

• \(m\) is the slope of the line.
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

This form is called the "slope-intercept form" and is widely used because it straightforwardly shows two critical features of a line: its slope and its intercept.

To find the equation from a line given two points, once the slope \(m\) is known, substitute \(m\) and one of the points into the equation form to solve for \(b\). This whole process allows us to predict and understand the behavior of the line simply by looking at its equation. Linear equations simplify complex relationships by providing a consistent and predictable framework for analyzing problems in algebra and calculus.