Problem 4 - Elementary Algebra

Question

Find the slope of the line determined by each pair of points. $$(-2,5),(-7,-1)$$

Step-by-Step Solution

Answer

The slope of the line through the points \((-2,5)\) and \((-7,-1)\) is \(\frac{6}{5}\).

Step 1: Understand the Slope Formula

The slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. This formula calculates the change in \( y \) values divided by the change in \( x \) values between the two points.

Step 2: Identify the Coordinates

Identify \( (x_1, y_1) \) and \( (x_2, y_2) \). For the points \((-2, 5)\) and \((-7, -1)\), let \((x_1, y_1) = (-2, 5)\) and \((x_2, y_2) = (-7, -1)\).

Step 3: Substitute the Values into the Slope Formula

Substitute the values from the points into the slope formula: \( m = \frac{-1 - 5}{-7 - (-2)}\).

Step 4: Simplify the Numerator

Calculate the change in \( y \): \(-1 - 5 = -6\). So the numerator is \(-6\).

Step 5: Simplify the Denominator

Calculate the change in \( x \): \(-7 - (-2) = -7 + 2 = -5\). So the denominator is \(-5\).

Step 6: Calculate the Slope

Now compute the slope: \[ m = \frac{-6}{-5} = \frac{6}{5} \]. Both the numerator and denominator are negative, which makes the slope positive.

Key Concepts

Algebra

In algebra, the concept of a slope plays a crucial role, especially when dealing with linear equations. The slope tells us how steep a line is and the direction it points.
When you deal with slopes:

• The slope is a measure of the vertical change relative to the horizontal change between two distinct points.
• This is often described as 'rise over run.'
• The slope provides insight into how fast one variable changes in relation to another.

For the points \[ (-2, 5) \] and \[ (-7, -1) \], algebra helps us identify change patterns by looking at differences in the coordinates. Recognizing these changes is the foundation of finding slopes. When you substitute these differences into the slope formula, a negative change in both x and y coordinates indicates a positive slope. This principle is fundamental in algebra for solving various real-life problems, like predicting trends.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebraic methods with geometric insight. When it comes to finding the slope of a line, coordinate geometry provides a visual understanding:

• Each point on a graph is represented by coordinates \( (x, y) \), which shows their specific location.
• The line connecting any two points on a coordinate plane can have its slope analyzed by understanding these coordinates.
• The slope can tell us whether the line moves in an upward or downward direction and its steepness.

For example, with points like \( (-2, 5) \) and \( (-7, -1) \), coordinate geometry allows us to plot these on the Cartesian plane. By understanding the change in y against the change in x, we can confirm if the line tilts upwards or downwards. This concept ensures a concrete visualization of abstract algebraic equations, making it an essential tool in mathematical problem-solving.

Mathematical Formula

Mathematical formulas are like recipes in cooking; they tell us exactly what ingredients (variables) we need and the steps to combine them. The slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]is the key to determining the slope of a line connecting any two points.

• This formula calculates the 'rise' (change in y-coordinates) over the 'run' (change in x-coordinates).
• Substituting values from your coordinate points \((-2, 5)\) and \((-7, -1)\) into the formula, you find the slope expresses the rate at which y changes with respect to x.
• Notably, a negative slope indicates a line moving downwards, while a positive one shows an upward direction.

Here, using the formula: \( m = \frac{-6}{-5} = \frac{6}{5} \), the result, \( \frac{6}{5} \), denotes a positive and moderate incline. Thus, the mathematical formula is an indispensable tool for translating geometric intuition into precise numerical expressions.