Problem 55 - College Algebra and Calculus: An Applied Approach

Question

Find an equation of the line passing through the points. \((-1,7),(3,7)\)

Step-by-Step Solution

Answer

The equation of the line passing through the points \((-1,7)\) and \((3,7)\) is \(y = 7\).

Step 1: Slope determination

The y-values for the given points \((-1,7)\) and \((3,7)\) are the same. This automatically implies that the line is horizontal, hence the slope \(m = 0\). In cases where the y-values differ, the slope is given by \((y2 - y1) / (x2 - x1)\) where \((x1,y1)\) and \((x2,y2)\) are the coordinates of the two points.

Step 2: Equation determination

The general formula for the equation of a line is \(y = mx + b\), where m is the slope and b is the y-intercept. Since we have a horizontal line, the equation simplifies to \(y = b\). The value of b can be determined by looking at the constant y-value from our points.

Step 3: Substituting for b

In examining the given points, the y-value of 7 is our constant. So when substituted into our simplified equation, we get: \(y = 7\).

Key Concepts

Slope Determination

When determining the slope of a line that passes through two given points, we first look at the coordinates. For the points \((-1,7)\) and \((3,7)\), the y-values of both points are the same. This tells us something very important: the line between these points is horizontal.
In the general case, if the y-values were different, we would calculate the slope, denoted as \(m\), using the formula:

• \(m = \frac{(y_2 - y_1)}{(x_2 - x_1)}\)

where \((x_1, y_1)\) and \((x_2, y_2)\) are your given points. When the y-values are the same, they cancel each other out, resulting in \(m = 0\).
This zero slope confirms that the line is indeed horizontal.

Horizontal Line

A horizontal line is a unique kind of line in mathematics because its slope is always zero. This is due to it not rising or falling as it moves along the x-axis. Hence, its y-values remain constant, which creates its flat characteristic.
For any set of points that lie on this line, the y-coordinate is identical no matter what the x-coordinate is. Because of this consistent y-value, the equation is straightforward:

• The equation becomes \(y = b\), where \(b\) is the constant y-value for all points on the line.

In our example, the y-value for both points \((-1,7)\) and \((3,7)\) is 7, making the equation of our horizontal line \(y = 7\). Understanding this leads us directly to finding the line's equation without additional computations.

Y-Intercept

The y-intercept of a line is the point where it crosses the y-axis. For most linear equations, the formula \(y = mx + b\) is used, where \(b\) denotes the y-intercept. However, with a horizontal line, the concept of y-intercept is slightly nuanced.
Since a horizontal line runs parallel to the x-axis, it doesn't actually "intersect" the y-axis like a typical sloped line does. Instead, the y-intercept \(b\) is simply the constant y-value across the entire line.
In cases like our example, where the line's equation is \(y = 7\), the 'intercept' isn't a point on the y-axis per se, but represents the value that y maintains throughout. So, \(b = 7\) is both the y-intercept and the value of y for all x on this horizontal line.