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Chapter 3: Problem 45

Find the slope of the line passing through the given points. See Examples 2 and 3 \((8,-4)\) and \((8,-3)\)

Short Answer

Expert verified
The slope is undefined, indicating a vertical line.

Step by step solution

01

Understand the Slope Formula

The formula for the slope of a line passing through two points (x_1, y_1) and (x_2, y_2) is given by \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where  m  represents the slope.
02

Substitute Point Values into the Formula

Here, the points given are (8, -4) and (8, -3). Substitute these points into the slope formula: \[ m = \frac{-3 - (-4)}{8 - 8} \] which simplifies to \[ m = \frac{-3 + 4}{8 - 8} \].
03

Simplify the Expression

Calculate the numerator: \(-3 + 4 = 1\). Thus, the slope formula becomes \[ m = \frac{1}{0} \].
04

Interpret the Result

Since we have a \(0\) in the denominator, the slope is undefined, indicating that the line is vertical.

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Key Concepts

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

Slope Formula
The slope formula is a crucial tool in understanding how steep or flat a line is on a coordinate plane. It tells us how much the line goes up or down vertically for every step it moves horizontally. The formula is written as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] In this formula, m refers to the slope. The letters x_1, y_1, x_2, and y_2 represent the x and y coordinates of any two points on the line. This formula helps us to:
  • Determine the tilt or slant of a line.
  • Know whether a line goes upwards or downwards.
  • Find out if a line is horizontal or vertical.
By plugging the points (8, -4) and (8, -3) into the formula, we observe that the x-values are the same. This is key and hints at a special type of line we'll uncover later.
Vertical Line
When examining lines on a graph, a vertical line is a line that goes straight up and down without any horizontal shift. This type of line is unique because its x-values remain constant, meaning all the points on the line have the same x-coordinate. Here are some characteristics of vertical lines:
  • Each point has the same x-coordinate.
  • There is no left or right movement.
  • Graphically, it appears as a up and down on the plane.
In our earlier example, both points (8, -4) and (8, -3) share the x-coordinate of 8. This identical x-value signifies a vertical line that does not change horizontally. Understanding vertical lines is essential for interpreting their slopes correctly.
Undefined Slope
An undefined slope occurs in the special case of a vertical line. In mathematical terms, it's when we try to use the slope formula, \[ m = \frac{y_2 - y_1}{x_2 - x_1} \], and end up dividing by zero. This happens with vertical lines because the denominator calculated as x_2 - x_1 is zero due to identical x-values. Let's break down the concept:
  • Dividing by zero in mathematics is non-allowable, hence the slope is undefined.
  • Vertical lines do not lean, so they cannot be described by a typical slope value.
In the case of points (8, -4) and (8, -3), using the slope formula gives us \frac{1}{0}, signaling that this line's slope is undefined. It's a helpful way to recognize and confirm the presence of a vertical line.

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Most popular questions from this chapter

For each pair of equations, determine whether their graphs are parallel, perpendicular, or neither. See Example 6 $$ \begin{aligned} &y=-2 x-9\\\ &2 x-y=9 \end{aligned} $$

Find the slope of a line perpendicular to the line passing through the given two points. See Example \(9 .\) \(\left(-2, \frac{1}{2}\right)\) and \(\left(-1, \frac{3}{2}\right)\)

Find the slope and the y-intercept of the graph of each equation and graph it. See Examples 4 and 5. $$ 2 x+3 y=9 $$

Find the slope and the y-intercept of the graph of each equation and graph it. See Examples 4 and 5. $$ y=\frac{x}{3} $$

For each pair of equations, determine whether their graphs are parallel, perpendicular, or neither. See Example 6 $$ \begin{aligned} &3 x=5 y-10\\\ &5 x=1-3 y \end{aligned} $$
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