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Magazine Chapter 5: Problem 84
Find the slope of the line that passes through each pair of points. $$ (6,7),(0,-5) $$
Short Answer
Expert verified
The slope of the line is 2.
Step by step solution
01
Identify the points
We are given two points through which the line passes: \((6,7)\) and \((0,-5)\). The first point, \((6,7)\), has coordinates \(x_1 = 6\) and \(y_1 = 7\). The second point, \((0,-5)\), has coordinates \(x_2 = 0\) and \(y_2 = -5\).
02
Use the slope formula
The formula for the slope \(m\) of a line passing 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
Substitute the coordinates of the points into the slope formula: \[ m = \frac{-5 - 7}{0 - 6} \].
04
Calculate the differences
Calculate the differences in the numerator and denominator: \(-5 - 7 = -12\) and \(0 - 6 = -6\). Thus, the formula becomes \[ m = \frac{-12}{-6} \].
05
Simplify the fraction
Simplify the fraction \(\frac{-12}{-6}\). Since both the numerator and denominator are negative, the signs cancel out, leaving us with \(\frac{12}{6} = 2\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is a branch of mathematics that utilizes a coordinate system to describe the position of points, lines, and figures in a plane. By combining algebra and geometry, it allows us to solve geometric problems using algebraic techniques.
- Points are defined by pairs of numbers called coordinates, which specify their position on the plane. The first number is the x-coordinate, and the second is the y-coordinate.
- Lines can be described by equations, and their properties such as slope can be analyzed using these equations.
- This powerful mathematical tool is essential for calculating distances, angles, and areas within the plane.
Slope Formula
The slope formula is a fundamental concept in coordinate geometry used to determine the steepness of a line. The slope of a line quantifies how much the y-coordinate of a point on the line changes per unit change in the x-coordinate. For two given points \(x_1, y_1\) and \(x_2, y_2\), the slope \(m\) is calculated using the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
- The numerator \(y_2 - y_1\) represents the vertical change or rise between the points.
- The denominator \(x_2 - x_1\) represents the horizontal change or run between the points.
- A positive slope indicates an upward trend, while a negative slope indicates a downward trend.
- A slope of zero means the line is horizontal, and an undefined slope occurs with a vertical line.
Linear Equations
A linear equation represents a straight line on the coordinate plane and can be expressed in the form \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept.
- The y-intercept is the point where the line crosses the y-axis.
- Linear equations are fundamental in describing relationships where one variable changes consistently with another.
- Solving linear equations involves finding the value of one variable based on the known value of another.
Coordinate Plane
The coordinate plane is a two-dimensional surface where each point is identified by a pair of numerical coordinates. These coordinates are typically written in the form \(x, y\).
- The x-axis is the horizontal line, and the y-axis is the vertical line. They intersect at the origin, \(0, 0\).
- Quadrants divide the plane into four regions, each offering unique characteristics based on the signs of their coordinates.
- Positive x and y coordinates are found in the upper-right quadrant, while negative coordinates populate the bottom-left.
