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Chapter 2: Problem 29

Find the slope of the line passing through the given points. \((-5,3)\) and \((-4,1)\)

Short Answer

Expert verified
The slope of the line is -2.

Step by step solution

01

Identify Points

We start by identifying the coordinates of the given points. The first point is \((-5, 3)\) and the second point is \((-4, 1)\).
02

Recall Slope Formula

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

Substitute Points into Formula

Substitute the coordinates of the points \((-5, 3)\) and \((-4, 1)\) into the slope formula:\[m = \frac{1 - 3}{-4 - (-5)}\]
04

Simplify the Expression

Simplify the expression to find the slope:\[m = \frac{-2}{-4 + 5} = \frac{-2}{1} = -2\]

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Key Concepts

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

Understanding Coordinates
When working with coordinates, you are basically identifying the exact position of a point on a graph. Imagine a graph with an x-axis (horizontal) and a y-axis (vertical). Coordinates are expressed as a pair \((x, y)\), where \(x\) is the value that moves you left or right, and \(y\) is the value that moves you up or down on the graph. Take the point \((-5, 3)\) for example:
  • The number \(-5\) tells you to move 5 units to the left.
  • The number \(3\) tells you to move 3 units up.
The key thing to remember is that coordinates need to be precise, as they determine where your points are on the grid. The same goes for the point \((-4, 1)\):
  • Move 4 units to the left from the origin, and 1 unit up.
By understanding coordinates, you can effectively plot points and analyze graphs in geometry or algebra.
Exploring the Slope Formula
The slope of a line is a measure of how steep the line is. To find the slope between two points, such as \((-5, 3)\) and \((-4, 1)\), we use the slope formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]In this formula:
  • \(m\) is the slope.
  • \(x_1\) and \(y_1\) represent the coordinates of the first point.
  • \(x_2\) and \(y_2\) represent the coordinates of the second point.
This formula is derived from the concept of rise over run. "Rise" is the change in the y-values, and "run" is the change in the x-values. By plugging the numbers from our points into the formula, we determine how much the line rises or falls as it moves from one point to another. Understanding the slope formula allows you to efficiently determine the inclination of any line.
Simplifying Expressions for the Slope
Simplifying expressions is a critical part of calculating the slope. After substituting the point values into the slope formula, you get an expression that needs to be simplified. Let's walk through the example \((-5, 3)\) and \((-4, 1)\) again:\[m = \frac{1 - 3}{-4 - (-5)}\]First, simplify the calculations in the numerator and denominator:
  • Numerator: \(1 - 3 = -2\)
  • Denominator: \(-4 - (-5)\) simplifies to \(-4 + 5 = 1\)
Then, compute the final simplified expression:
  • Divide \(-2\) by \(1\) to get the slope:\[m = \frac{-2}{1} = -2\]
Simplifying expressions ensures that you arrive at the simplest form of your answer, making calculations easy to manage and results quick to interpret.

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