Problem 12 - Algebra and Trigonometry

Question

Find the slope of the line through P and Q. $$ P(-1,-4), Q(6,0) $$

Step-by-Step Solution

Answer

The slope of the line is \( \frac{4}{7} \).

Step 1: Identify Coordinates

First, identify the coordinates of the points given in the problem. The point \( P \) has coordinates \((-1, -4)\), while the point \( Q \) has coordinates \((6, 0)\). These are the coordinates that we will use to calculate the slope of the line passing through these points.

Step 2: Apply Slope Formula

Use the slope formula to determine the slope of the line. The formula is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula helps us find the rate of change between any two points on a line.

Step 3: Substitute the Coordinates

Substitute the coordinates of the points \( P(-1, -4) \) and \( Q(6, 0) \) into the slope formula. Let \( (x_1, y_1) = (-1, -4) \) and \( (x_2, y_2) = (6, 0) \). The formula becomes: \[ m = \frac{0 - (-4)}{6 - (-1)} \].

Step 4: Calculate the Numerator

Calculate the difference in the y-coordinates (numerator of the slope formula): \(0 - (-4) = 0 + 4 = 4\).

Step 5: Calculate the Denominator

Calculate the difference in the x-coordinates (denominator of the slope formula): \(6 - (-1) = 6 + 1 = 7\).

Step 6: Divide Numerator by Denominator

Finally, divide the result of the numerator by the result of the denominator to find the slope: \[ m = \frac{4}{7} \]. This is the slope of the line through points \( P \) and \( Q \).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using algebraic concepts to understand geometrical concepts on a coordinate plane. Unlocking the mysteries of the coordinate plane helps us visualize mathematical equations more concretely.

On the coordinate plane, you'll encounter two main axes—horizontal (x-axis) and vertical (y-axis). Each point is pinpointed by an ordered pair of numbers known as coordinates

• The first number, known as the x-coordinate, tells you how far to move horizontally from the origin.
• The second number, the y-coordinate, gives you the vertical movement.

The beauty of coordinate geometry is its power to describe all kinds of lines and shapes simply through these coordinates. When given points like \( P(-1,-4) \) and \( Q(6,0) \), they become markers on the grid, helping us understand relative positions and calculate properties like distance and slope.

Rate of Change

The rate of change is a fundamental concept that describes how one quantity changes in relation to another. In the context of a line, it specifically refers to the change in the y-coordinate with respect to the x-coordinate.

This concept is visually represented by the slope of a line, which shows how steep or flat a line is as it stretches across the coordinate plane.

• A positive slope indicates that as x increases, y increases. Imagine a line rising as it moves from left to right.
• A negative slope means that as x goes up, y goes down—picture a line descending from left to right.
• Zero slope implies a perfectly horizontal line, showing no change in y over x.
• An undefined slope is characteristic of a vertical line where x doesn’t change.

The rate of change is pivotal in numerous fields, like physics and economics, indicating how systems evolve over time or conditions. For our specific line through points \( P \) and \( Q \), the rate of change is calculated to be \( \frac{4}{7} \), suggesting that for every 7 units of horizontal change, y increases by 4 units.

Slope Formula

The slope formula is a mathematical tool used to determine the steepness and direction of a line on a graph. It is expressed as \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where

• \(m\) represents the slope,
• \(x_1, y_1\) are the coordinates of the first point,
• \(x_2, y_2\) are the coordinates of the second point.

This formula is your go-to for finding out how much y changes for a unit change in x, essentially telling us the rate of the line's elevation. Calculating slope helps assess whether a line ascends, descends, or remains level.

Using this formula requires you to identify the changes in y and x coordinates separately, as demonstrated with our points \( P(-1,-4) \) and \( Q(6,0) \):

• Change in y (\(y_2 - y_1\)) is \(0 + 4 = 4\).
• Change in x (\(x_2 - x_1\)) is \(6 + 1 = 7\).

Thus, the slope \( \frac{4}{7} \) describes how these coordinates translate into a direct and constant line pattern on the coordinate plane.