Problem 31 - College Algebra with Corequisite Support

Question

For the following exercises, find the equation of the line using the given information. (1,7) and (3,7)

Step-by-Step Solution

Answer

The equation of the line is \(y = 7\).

Step 1: Identify the Point Coordinates

We are given two points: \((1, 7)\) and \((3, 7)\). From these, we identify the coordinates, where the first point \((x_1, y_1)\) is \((1, 7)\) and the second point \((x_2, y_2)\) is \((3, 7)\).

Step 2: Calculate the Slope (m)

To find the slope \(m\) of the line, we use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the given values, \(y_2 = 7\), \(y_1 = 7\), \(x_2 = 3\), and \(x_1 = 1\), we have: \[ m = \frac{7 - 7}{3 - 1} = \frac{0}{2} = 0 \] This indicates the slope is \(0\), meaning the line is horizontal.

Step 3: Write the Equation of the Line

For a line with a slope of \(0\), the equation follows the form \(y = b\) where \(b\) is the constant \(y\)-value for any \(x\). Given that both points share the \(y\)-coordinate of \(7\), the equation is \(y = 7\).

Key Concepts

Slope of a Line

The slope of a line is a measure of its steepness or incline. In mathematics, the slope is usually represented by the letter \( m \). The formula to calculate the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This tells us how much \( y \) changes for a unit change in \( x \).

• If the slope is positive, the line rises as it moves from left to right.
• A negative slope indicates the line falls as it moves from left to right.
• A slope of zero means the line is perfectly horizontal.

The slope is a crucial concept because it defines the angle and direction of the line on a graph. Once you know the slope, you have a lot of information about the line's tilt and orientation.

Coordinates

Coordinates are a way of locating points in a plane using pairs of numbers. Each point on a Cartesian plane is identified by an \((x, y)\) pair known as its coordinates.

• The first number \(x\) is the horizontal position (left or right).
• The second number \(y\) is the vertical position (up or down).

Coordinates give us a precise way to plot and read points on a graph. For example, in this exercise, the points (1, 7) and (3, 7) tell us that their horizontal placements differ, but they share the same vertical position. Knowing these points helps us determine the slope and equation of the line that connects them.

Horizontal Line

A horizontal line is a straight line that runs left to right across the graph. Its most distinct characteristic is that it has a slope of zero. This means there's no vertical change as you move down the line, only horizontal change. For a horizontal line, the equation takes the form \( y = b \), where \( b \) is some constant. This means no matter what \( x \)-value you choose, \( y \) stays the same. In the exercise, since the points given are (1, 7) and (3, 7), the \( y \)-value for both points is 7. Thus, the line is horizontal, and the equation of the line is \( y = 7 \). Horizontal lines are easy to identify and plot because they maintain a consistent height across the graph.