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Chapter 15: Problem 24

Find the slope of the line containing each given pair of points. \((-1,7)\) and \((-3,4)\)

Short Answer

Expert verified
The slope is \( \frac{3}{2} \).

Step by step solution

01

- Identify the coordinates

Identify the given coordinates. Point 1: \((-1, 7)\), Point 2: \((-3, 4)\)
02

- Write down the slope formula

The formula to find the slope (m) of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
03

- Substitute the coordinates into the slope formula

Substitute \((-1, 7)\) and \((-3, 4)\) into the slope formula:\[ m = \frac{4 - 7}{-3 + 1} = \frac{-3}{-2} \]
04

- Simplify the fraction

Simplify the fraction obtained to find the slope:\[ m = \frac{-3}{-2} = \frac{3}{2} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

slope of a line
The slope of a line is a measure of its steepness and direction.
It tells us how much the y-coordinate of a point changes for each change in the x-coordinate. If the slope is positive, the line rises as it moves from left to right. If the slope is negative, the line falls.
Think of the slope as the 'rise' over the 'run'.
  • A slope of 0 means the line is horizontal.
  • An undefined slope means the line is vertical.
For our given points, understanding the rise and run helps us visualize how the line behaves in a coordinate plane.

slope formula
The slope formula is crucial in finding the slope from two points.
For points \((x_1, y_1)\) and \((x_2, y_2)\), the formula to find the slope, denoted as \ m \, is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This formula gives the difference in y-coordinates (rise) divided by the difference in x-coordinates (run).
In our exercise, substituting \((-1, 7)\) and \((-3, 4)\) into the formula gives us a clear path to find the slope.
coordinate geometry
Coordinate geometry, also called analytic geometry, is the study of geometry using a coordinate system.
It helps us to use algebra to find geometric properties, such as slopes of lines, distances between points, and shapes of graphs.
In our exercise, coordinate geometry allows us to plot points \((-1, 7)\) and \((-3, 4)\) on a graph and use the slope formula to determine the line's slope directly from the graph. Using this approach simplifies many aspects of geometry by using algebraic techniques.
fraction simplification
Fraction simplification is the process of reducing a fraction to its simplest form.
For any fraction \frac{a}{b}\, we can simplify by dividing both the numerator and the denominator by their greatest common divisor (GCD).
For example, in our exercise: \[ m = \frac{-3}{-2} = \frac{3}{2} \] We see that dividing negative signs in the numerator and denominator gives us a positive fraction. Always look to simplify fractions to their simplest form to make calculations easier and results clearer.

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