Problem 1 - The Calculus with Analytic Geometry

Question

In Exercises 1 through 4 , find the slope of the line through the given points. $$ (2,-3),(-4,3) $$

Step-by-Step Solution

Answer

The slope is -1.

Step 1: Understand the Slope Formula

The formula to find the slope (m) between two points \(x_1, y_1\) and \(x_2, y_2\) 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 the two points.

Step 2: Identify the Coordinates

Identify the coordinates of the given points: \(x_1 = 2, y_1 = -3\) and \(x_2 = -4, y_2 = 3\).

Step 3: Substitute the Coordinates into the Formula

Substitute the values into the slope formula: \[ m = \frac{3 - (-3)}{-4 - 2} \]

Step 4: Simplify the Expression

Simplify the numerator and the denominator: \[ m = \frac{3 + 3}{-4 - 2} = \frac{6}{-6} = -1 \]

Step 5: Interpret the Result

The slope \(m\) of the line passing through the points (2, -3) and (-4, 3) is -1.

Key Concepts

Slope Formula

The slope of a line measures its steepness and direction. It's an important concept in algebra and coordinate geometry, often referred to as the 'rise over run.' To find the slope, we often use the slope formula:

• The slope formula is given by \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]


• Here, \(x_1, y_1\) and \(x_2, y_2\) are the coordinates of two distinct points on the line.

Using this formula, you can calculate how much the y-coordinate (vertical change) changes for each unit increase in the x-coordinate (horizontal change). Always make sure to subtract the coordinates in the same order to avoid mistakes. If you mix the coordinates or the order, the slope calculation will be incorrect. Keep practicing with different pairs of points to get a good grip on the slope formula.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system.

Imagine the x-axis and y-axis on a plane, which help in locating points called coordinates. The basics involve understanding how points, lines, and shapes are positioned and how they relate to each other. Here are some key points:

• The coordinates (or ordered pairs) are written as \(x, y\).
• The x-coordinate is the location of the point along the horizontal axis.
• The y-coordinate is the position along the vertical axis.


For example,
The point (2, -3) lies 2 units to the right and 3 units downward from the origin (0,0). Similarly, the point (-4, 3) is 4 units left and 3 units up from the origin. Understanding coordinate geometry is fundamental to solving problems like finding the slope of a line. It helps you visualize and interpret the spatial relationships between different elements on the plane.

Linear Equations

Linear equations describe straight lines in coordinate geometry. These are algebraic equations that have a constant rate of change and can be written in several forms:

• Standard Form: \[ Ax + By = C \]
• Slope-Intercept Form: \[ y = mx + b \]

In these forms, m represents the slope, and b is the y-intercept (where the line crosses the y-axis). If you know the slope of a line and a point on the line, you can write the equation of the line. For example, with a slope of -1 and knowing it passes through (2,-3), you could derive its linear equation. Linear equations send a clear message: the change between any two points is constant. This consistency helps in forming a solid understanding of lines and their behavior on a plane.

Point-Slope Form

The point-slope form of a linear equation is particularly useful when you know the slope and one point on the line. It is written as: \[ y - y_1 = m(x - x_1) \]
Here:

• \(m\) is the slope


• \(x_1, y_1\) is a point on the line

Starting with the slope -1 and a point, for instance, (2,-3), you can use the point-slope formula to find the linear equation.Substituting \(m = -1\) and \(x_1 = 2, y_1 = -3\):\[ y - (-3) = -1(x - 2) \]

By simplifying, you get
\[ y + 3 = -1(x - 2)\]Then, distribute and rearrange to standard or slope-intercept form, depending on what you need. Point-slope form makes it easier to write equations when data is limited and clear understanding of these equations will ease up tasks of finding lines equations straightforwardly.