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Chapter 1: Problem 12

Find the equation of the line that passes through the two given points. Write the line in slope-intercept form \((y=m x+c)\), if possible. $$\left(\frac{7}{3},-\frac{1}{2}\right) \text { and }\left(\frac{7}{3}, \frac{5}{2}\right)$$

Short Answer

Expert verified
The line is vertical with equation \( x = \frac{7}{3} \).

Step by step solution

01

Find the Slope (m)

The first step is to calculate the slope of the line passing through two points: \( \left(\frac{7}{3}, -\frac{1}{2}\right) \) and \( \left(\frac{7}{3}, \frac{5}{2}\right) \). The slope formula is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).Here, substitute the given points into the slope formula: \[ m = \frac{\frac{5}{2} - \left(-\frac{1}{2}\right)}{\frac{7}{3} - \frac{7}{3}} \]This simplifies to:\[ m = \frac{\frac{5}{2} + \frac{1}{2}}{0} = \frac{6}{2} \times \frac{1}{0} = \text{undefined} \]
02

Identify the Line Type

Since the slope \( m \) is undefined, the line is vertical. A vertical line has a constant \( x \) value for all \( y \), which is based on the given points.
03

Write the Equation of the Vertical Line

For vertical lines, the equation is \( x = c \), where \( c \) is the constant \( x \) coordinate from the given points. In this case, \( x = \frac{7}{3} \).
04

(Bonus): Verify if Slope-Intercept Form is Possible

The slope-intercept form \( y = mx + c \) is not possible for vertical lines because the line has an undefined slope and does not functionally relate \( y \) to \( x \) with a single-valued function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope Calculation
The slope of a line tells us how steep the line is, essentially the "rise" over the "run." It measures the change in the y-coordinate as the x-coordinate changes. To find the slope, we use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two distinct points on the line. When calculating the slope, it鈥檚 important to remember:
  • Subtract the y-values and x-values respectively.
  • If the difference in x-values \((x_2 - x_1)\) is zero, the slope is undefined.
In our exercise, the slope turns out to be \( \frac{6}{2} \times \frac{1}{0} \), which is undefined, indicating a vertical line!
Vertical Line
A vertical line is unique in the fact that it travels straight up and down across a graph. Unlike most lines, its slope is undefined because there鈥檚 no horizontal change (the "run"), just vertical movement (the "rise").Key characteristics of vertical lines:
  • They have an equation form of \( x = c \), where \( c \) is constant.
  • Every point on a vertical line has the same x-coordinate, making it unique compared to other lines.
  • Slope-intercept form \( y = mx + c \) does not work for vertical lines, since they can't express \( y \) as a function of \( x \).
In our particular exercise, since both given points have the same x-coordinate \( \left(\frac{7}{3}\right)\), we identify the line as vertical: \( x = \frac{7}{3} \).
Equation of a Line
Writing an equation of a line generally involves understanding its structure and type. For most lines, the slope-intercept form \( y = mx + c \) fits well, where \( m \) is the slope and \( c \) is the y-intercept.For vertical lines, however, things differ:
  • Since their slope is undefined, they cannot be represented in slope-intercept form.
  • The equation instead remains \( x = c \), representing a constant x-value.
Understanding the equation of a vertical line helps to visualize that all points on the line share the same x-coordinate, aligned straight up or down along the x-axis. It鈥檚 an easy way to identify the line without dealing with y-values.

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Most popular questions from this chapter

Solve the inequality. $$9 \leq 8 x-3<11$$

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