Problem 54 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write an equation of the line that passes through the points. Use the slope- intercept form (if possible). If not possible, explain why and use the general form. Use a graphing utility to graph the line (if possible). $$(-1,6),(5,6)$$

Step-by-Step Solution

Answer

The equation of the line that passes through the points (-1,6) and (5,6) is \(y = 6\).

Step 1: Compute the Slope

First, using the two given points (-1,6) and (5,6), compute the slope of the line using the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]. Substituting the given points into the formula, it gives \(m = \frac{6 - 6}{5 - (-1)} = 0\).

Step 2: Formulate the Equation

Since the slope \(m = 0\), according to the slope-intercept equation, \(y = mx + b\), it means the line is a horizontal line, and it passes through the y-coordinate 6, so the equation would be \(y = 6\).

Step 3: Plot the Graph

Using a graphing utility, plot the line using the equation \(y = 6\). Since it's a horizontal line, it would run parallel to the x-axis through the point (0,6). Any pair of x, y coordinates where y equals 6 will be on this line.

Key Concepts

Slope-Intercept Form

When discussing the equation of a line, the slope-intercept form is perhaps one of the most convenient and popular representations. It's written as
\( y = mx + b \)

where \( m \) is the slope of the line—essentially a measure of its steepness—and \( b \) is the y-intercept, the point where the line crosses the y-axis. These two pieces of information are enough to draw the entire line on a graph! For a beginning student, this form is incredibly useful because it makes two things clear instantly: the direction and the