Problem 28 - College Algebra

Question

For the following exercises, find the slope of the line that passes through the two given points. $$ (6,11) \text { and }(-4,3) $$

Step-by-Step Solution

Answer

The slope is \( m = \frac{4}{5} \).

Step 1: Understanding the Formula for Slope

The slope \( m \) of a line going through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula finds the change in \( y \) over the change in \( x \), also known as 'rise over run'.

Step 2: Identify the Points

Identify the values of your points: \((x_1, y_1) = (6,11)\) and \((x_2, y_2) = (-4,3)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics where algebraic equations are used to represent geometric shapes and curves. It allows us to express geometric properties and relationships using coordinates on a plane. Think of it like using a map where each location or point has coordinates that tell you exactly where it is.

In a Cartesian coordinate system, which is most commonly used in coordinate geometry, every point is identified by an x-coordinate and a y-coordinate. These are written as (x,y).

• The x-coordinate tells you how far to go horizontally.
• The y-coordinate tells you how far to go vertically.

This allows us to place points, and eventually graphs of equations, on a two-dimensional plane.

Coordinate geometry is the backbone for understanding how lines, circles, and other shapes are drawn and analyzed mathematically. It's essential for calculating distances, finding intersections, and understanding the idea of 'slopes' of lines.

Calculating Slopes

Calculating the slope of a line is a fundamental concept in coordinate geometry. It allows us to understand how steep a line is, as well as its direction. The slope, often denoted by the letter m, can be thought of as the 'tilt' of a line on a graph.

To calculate the slope between two points (x₁, y₁) and (x₂, y₂), you use the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
This formula expresses the idea of "rise over run," which means:

• Rise - the vertical change or the difference between the y coordinates (y₂ - y₁).
• Run - the horizontal change or the difference between the x coordinates (x₂ - x₁).

These differences tell you how much and in which direction the line moves from one point to the other.

For example, using the formula for points (6, 11) and (-4, 3), we find:
\[ m = \frac{3 - 11}{-4 - 6} = \frac{-8}{-10} = \frac{4}{5} \]
This slope indicates that for every 5 units you move horizontally, the line rises 4 units vertically.

Linear Equations

Linear equations are mathematical statements that describe straight lines. They are crucial for understanding relationships between two quantities that change proportionally. They usually have a general form where you're tasked with finding the slope and y-intercept.

A common form of a linear equation is the slope-intercept form, given by:
\[ y = mx + b \]
In this equation:

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

This form clearly shows how the line behaves as x changes and makes predicting other points on the line straightforward.

For instance, if a line has a slope of \(\frac{4}{5}\) (rises 4 units for every 5 units it moves horizontally) and crosses the y-axis at 2 (b = 2), the equation becomes:
\[ y = \frac{4}{5}x + 2 \]
Understanding linear equations helps in modeling and solving problems related to rates, trends, and predictions.