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Chapter 2: Problem 27

Find the slope of the line that passes through the given points, if possible. See Example 2. $$ (-1,8),(6,1) $$

Short Answer

Expert verified
The slope of the line is -1.

Step by step solution

01

Understand the Slope Formula

The slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].
02

Identify Given Points

Identify the coordinates of the given points. Here, \( x_1 = -1 \), \( y_1 = 8 \), \( x_2 = 6 \), and \( y_2 = 1 \).
03

Substitute Values into Slope Formula

Substitute the identified values into the slope formula: \[ m = \frac{1 - 8}{6 - (-1)} \].
04

Simplify the Expression

Simplify the expression in the slope formula: \[ m = \frac{1 - 8}{6 + 1} = \frac{-7}{7} \].
05

Calculate the Final Slope

Simplify \( \frac{-7}{7} \) to obtain the final slope: \( m = -1 \).

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Key Concepts

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

Slope Formula
The slope formula is crucial when determining how steep a line is, using two given points. Think of the slope as a measure of how much a line rises vertically for every unit it moves horizontally. The formula is:
  • \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
Here, \( m \) represents the slope. The values \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points through which the line passes. By plugging in the appropriate values, you can quickly find the slope. Just remember, the key is having two distinct points to avoid division by zero.
Point Coordinates
Point coordinates are sets of numerical values that define the position of a point in space. In two-dimensional space, coordinates are given as a pair \((x, y)\).
For example, in our exercise, we are given the points \((-1, 8)\) and \((6, 1)\). Each pair contains an \(x\)-coordinate and a \(y\)-coordinate, respectively.
  • \((-1, 8)\) has \(x = -1\) and \(y = 8\).
  • \((6, 1)\) has \(x = 6\) and \(y = 1\).
Knowing how to read and apply point coordinates allows us to use them in formulas such as the slope formula to discover more information about geometric figures like lines.
Line Equations
Line equations provide a method to represent any line on a plane, typically expressed in the form \( y = mx + b \). Here, \( m \) is the slope and \( b \) is the y-intercept鈥攖he point where the line crosses the y-axis.
By using the slope from our earlier calculation \( m = -1 \), we can write the equation of the line if we know the y-intercept.
Sometimes the equation is reorganized, but the main idea remains the same: to express the relationship between x and the corresponding value of y using constant values.
Simplifying Fractions
Simplifying fractions is a vital step in obtaining the neatest form of a mathematical expression. It involves reducing a fraction to its simplest equivalent where the numerator and the denominator have no common factors other than 1.
In our exercise, we calculated the slope as \( \frac{-7}{7} \), which can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 7, giving us \( -1 \).
  • Divide \(-7\) by 7 = \(-1\)
  • Divide 7 by 7 = 1
The result \( m = -1 \) means we've successfully simplified the fraction, perfectly illustrating why simplifying is a powerful tool for clarity and utility.

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