Problem 19 - Understanding Elementary Algebra with Geometry

Question

Find the slope of the line passing through the given points. Round to the nearest hundredth where necessary. \((-1,3)\) and \((3,-1)\)

Step-by-Step Solution

Answer

The slope is -1.

Step 1: Identify the points

We are given two points: \( -1, 3 \) and \( 3, -1 \). Let's denote the first point as \( (x_1, y_1) \) and the second point as \( (x_2, y_2) \). That gives us \( x_1 = -1 \), \( y_1 = 3 \), \( x_2 = 3 \), \( y_2 = -1 \).

Step 2: Use the slope formula

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

Step 3: Substitute the values

Substituting the given values into the slope formula: \[ m = \frac{-1 - 3}{3 - (-1)} = \frac{-4}{4} \]

Step 4: Simplify the expression

Simplify \( \frac{-4}{4} \) to get: \[ m = -1 \]

Key Concepts

Slope Formula

The slope of a line is a number that describes both the direction and the steepness of the line. The slope formula, an essential concept in coordinate geometry, is often represented by the letter \( m \). It is calculated using the formula:

• The difference in the y-coordinates of two points


• Divided by the difference in the x-coordinates of the same points

If given points \( (x_1, y_1) \) and \( (x_2, y_2) \), the formula for the slope \( m \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
This formula tells us how much the line rises or falls vertically for each horizontal unit it moves. To find the slope:

• Identify the coordinates of the two points.


• Plug these values into the slope formula.


• Simplify to find the slope.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to use algebra to solve geometric problems. It involves using a coordinate system to represent geometric shapes and analyze their properties. In a coordinate plane:

• The horizontal line is called the x-axis.


• The vertical line is called the y-axis.


• Points are represented as \( (x, y) \) where \( x \) is the x-coordinate and \( y \) is the y-coordinate.

The slope of a line is particularly relevant in coordinate geometry because it helps determine the angle and direction of the line. By knowing the coordinates of two points, such as \( (-1, 3) \) and \( (3, -1) \), we can apply the slope formula to understand the steepness and orientation of the line connecting these points.

• This information is valuable for graphing lines.


• It helps in writing the equations of lines.


• It’s used in finding the relationship between different lines.

Linear Equations

Linear equations are equations of the first order. They describe a straight line when graphed on a coordinate plane. The general form of a linear equation is:
\[ y = mx + b \]
Where:

• \( y \) represents the y-coordinate.


• \( m \) is the slope of the line.


• \( x \) is the x-coordinate.


• \( b \) is the y-intercept (the point where the line crosses the y-axis).

Finding the slope using points is a fundamental step in determining the equation of the line. Once you have the slope \( m \), you can use one of the points and solve for \( b \). For example, with points \( (-1, 3) \) and \( (3, -1) \) and slope \( m = -1 \):

• Use the point-slope form \( y - y_1 = m(x - x_1) \).


• Substitute one set of coordinates and the slope.


• Simplify to the form \( y = mx + b \).

This process allows you to find the complete equation of the line, making it easier to graph and analyze.