Problem 29 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find an equation of the line that passes through the given point and has the indicated slope. Sketch the line by hand. Use a graphing utility to verify your sketch, if possible. \((6,-1), \quad m\) is undefined.

Step-by-Step Solution

Answer

The equation of the line which passes through the point (6,-1) and has an undefined slope is \(x = 6\).

Step 1: Understand the concept of undefined slope

An undefined slope is associated with a line that runs vertically. This means its slope is undefined because the run (change in x) is zero, and you can't divide by zero.

Step 2: Formulate Equation of the Line

As we know, the equation for a vertical line is \(x = a\), where \(a\) is the x-coordinate of a point on the line. Here, our given point is (6, -1). So, replacing \(x\) with \(6\), we get \(x = 6\).

Step 3: Verify line with Graphing Utility

If you plot a line with \(x = 6\) using any graphing utility, the line would be vertical through \(x = 6\), which aligns with the initial description.

Key Concepts

Equation of a Line

The equation of a line is a mathematical expression that describes all the points that lie on that line. The most frequently used form is the slope-intercept form, expressed as \[ y = mx + b \] where

• \( m \) is the slope,
• \( b \) is the y-intercept of the line.

This form is handy when you know the slope and y-intercept. However, when dealing with vertical lines, the slope \( m \) is undefined.
For vertical lines, the change in the x-coordinate is zero, meaning you cannot write this line in slope-intercept form. Instead, it is written in the format \[ x = a \]where \( a \) is a constant. This contrasts with non-vertical lines, which have a defined slope.

Vertical Line

A vertical line is a special type of line in geometry that extends infinitely in the vertical direction. The identifying feature of a vertical line is that all points on the line share the same x-coordinate.
In the case of our exercise, the point \( (6, -1) \) indicates that the line we are dealing with must have an undefined slope. This happens because there is no horizontal movement (or change) as you move along the line, which means you are not traveling left or right.
Hence, such a line is mathematically represented by the equation \( x = 6 \).Every point on this line will have an x-coordinate of 6 and varying y-coordinates depending on how far up or down you go.

Graphing Utility

Graphing utilities are tools such as calculators or software applications used to graph mathematical equations. They provide an easy visual representation of equations on a Cartesian plane, allowing users to better understand the graphical relationship between variables.
When examining the vertical line defined by our equation \( x = 6 \), using a graphing utility verifies the characteristics of the line. You would enter this equation into the graphing utility, and it would display a straight line crossing the x-axis at 6. This visual confirmation is invaluable for ensuring calculations are done correctly and the line behaves as expected.
With modern technology, many graphing utilities can also provide additional features, like calculating slopes, finding intersections, and more, enhancing their educational utility.