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Chapter 2: Problem 65

Find the equation of the line that passes through the following points: \((a, b)\) and \((a, b+1)\)

Short Answer

Expert verified
The line is vertical, undefined slope.

Step by step solution

01

Identify Point Coordinates

We have two points: \((a, b)\) and \((a, b+1)\). These points will help us determine the slope and equation of the line.
02

Find the Slope of the Line

The formula for the slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). With our points \((a, b)\) and \((a, b+1)\), the slope becomes:\[ m = \frac{(b+1) - b}{a - a} = \frac{1}{0} \]

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Key Concepts

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

Slope of a Line
The slope of a line is a measure of how steep or flat the line is. It tells us how the y-value changes for a unit change in the x-value. To find the slope, we use the formula:
  • \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
For example, if we have two points, \((x_1, y_1)\) and \((x_2, y_2)\), the slope \(m\) can be calculated using the differences in their y-coordinates and x-coordinates.
In the exercise, we have the points \((a, b)\) and \((a, b+1)\). Let's plug them into the slope formula. Substitute these points into the formula:
  • \(m = \frac{(b+1) - b}{a - a} = \frac{1}{0}\)
Here, you see an important situation arises. There is division by zero, which leads to the concept of an undefined slope.
Equation of a Line
Once we know the slope, we can find the equation of the line. The equation of a line in slope-intercept form is:
  • \(y = mx + c\)
\(m\) is the slope, and \(c\) is the y-intercept.
However, when we encounter special cases like vertical lines, the approach differs slightly.
  • Vertical lines arise when a line passes through points with the same x-coordinate.
  • Such lines are parallel to the y-axis.
In our exercise, both points \((a, b)\) and \((a, b+1)\) have the same x-coordinate \(a\). Thus, instead of using the slope-intercept form, which isn't possible due to an undefined slope, the line's equation simplifies to:
  • \(x = a\)
Here, \(x = a\) represents a vertical line.
Undefined Slope
An undefined slope occurs when the denominator of the slope formula becomes zero. This happens when the x-values of the two points on the line are the same, leading to
  • \(x_2 - x_1 = 0\)
  • \(\frac{1}{0}\) is undefined in mathematics.
An undefined slope indicates a vertical line.
Vertical lines are unique because they do not follow the typical linear pattern. They do not have a defined slope as you'd see in lines that rise or fall from left to right.
  • Their representation remains constant at \(x = \) some value.
Understanding undefined slopes is crucial in distinguishing between vertical and non-vertical lines. In practical terms, for any pair of points where the x-values remain identical, recognize it as a vertical line with an equation \(x = \) constant.

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