Problem 8 - College Algebra with Modeling and Visualization

Question

If possible, find the slope of the line passing through each pair of points. $$ (-8,5),(-3,-7) $$

Step-by-Step Solution

Answer

The slope is \(-\frac{12}{5}\).

Step 1: Understanding the Slope Formula

The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Where \(m\) is the slope.

Step 2: Plug in the Coordinates

Identify the first point \((x_1, y_1) = (-8, 5)\) and the second point \((x_2, y_2) = (-3, -7)\), then substitute these into the slope formula:\[m = \frac{-7 - 5}{-3 - (-8)}\]

Step 3: Simplify the Numerator

Calculate the difference in the y-coordinates:\(-7 - 5 = -12\).

Step 4: Simplify the Denominator

Calculate the difference in the x-coordinates:\(-3 - (-8) = -3 + 8 = 5\).

Step 5: Calculate the Slope

Substitute the simplified numerator and denominator back into the slope formula:\[m = \frac{-12}{5}\]Thus, the slope of the line is \(-\frac{12}{5}\).

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of mathematics that deals with the study of geometric figures through the use of coordinates on a plane. Imagine a huge graph paper where every point has a unique address given by two numbers. These numbers are the x (horizontal) and y (vertical) coordinates. This system is especially useful because it allows us to analyze complex shapes and lines using algebra.
Coordinates give us precise locations, making it possible to specify exactly where points are in relation to one another. This system underlies much of modern mathematics and physics, providing a basis for understanding distances, angles, and shapes.

• For example, the point \((-8, 5)\) tells us its exact location: 8 units to the left and 5 units up from the origin \(0, 0\).
• Similarly, \((-3, -7)\) is positioned 3 units to the left and 7 units down.

By plotting points and considering lines between them, coordinate geometry opens the pathway to understanding slopes, distances, and equations of lines, bringing mathematics into the tangible world.

Slope Formula

The slope formula is at the heart of understanding lines in coordinate geometry. When two points are known, the slope helps quantify how steep the line connecting them is. Slope essentially measures the 'rise' over the 'run', which means vertically down or up for every horizontal unit moved to the right. In mathematical terms, this relationship is expressed as follows: \(m = \frac{{y_2-y_1}}{{x_2-x_1}}\).
This formula is crucial because it provides a simple way to characterize and compare lines.
The task of finding the slope is like finding out how much you climb or descend when you walk from one point to another along a line. Using our points from the exercise:

• Point 1: \((-8, 5)\)
• Point 2: \((-3, -7)\)

We see that substituting these coordinates into the formula helps determine the slope efficiently. Let's break it down next.

Mathematics Problem Solving

Mathematics problem solving is both an art and a science. It requires employing logical steps to arrive at a solution. For our exercise, solving for the slope involves clear steps:

• Identify and label the given points: Point 1 \(((x_1, y_1) = (-8, 5)\)) and Point 2 \(((x_2, y_2) = (-3, -7)\))
• Apply the numbers to the slope formula and calculate: \(m = \frac{{-7-5}}{{-3-(-8)}}\).
• Simplify each part to find the final slope: \(-12\) in the numerator and \(+5\) in the denominator, leading to \(-\frac{12}{5}\).

Step-by-step simplification allows for accuracy and understanding. Each part of the slope calculation builds upon the last, emphasizing the importance of careful and deliberate calculations.
Through this exercise, we gain not only a number — the slope \(-\frac{12}{5}\) — but also a deeper appreciation of how points interact in space. This clarity and precision are what make problem solving an essential skill in mathematics.