Problem 40 - Algebra 2

Question

Find the slope of the line that passes through each pair of points. $$ (c, 5),(c,-2) $$

Step-by-Step Solution

Answer

The slope of the line is undefined.

Step 1: Understanding the Formula for Slope

The slope (m) of a line that passes through two points \(x_1, y_1\) and \(x_2, y_2\) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] We will use this formula to find the slope of the line through the given points.

Step 2: Identify the Coordinates

The given points are \(c, 5\) and \(c, -2\). Here, \((x_1, y_1) = (c, 5)\) and \((x_2, y_2) = (c, -2)\). Notice that both points have the same x-coordinate (c).

Step 3: Substitute into the Slope Formula

Substitute the identified coordinates into the slope formula: \[ m = \frac{-2 - 5}{c - c} \] which simplifies to \[ m = \frac{-7}{0} \]

Step 4: Evaluate the Slope

When the denominator of the slope formula is 0, the slope is undefined because division by zero is not possible. In this case, since the x-coordinates are the same, the line is vertical.

Key Concepts

Vertical Line

In geometry, a vertical line is a line that runs straight up and down on a coordinate plane. It has a unique characteristic: every point on a vertical line shares the same x-coordinate. For example, if we have two points

• Point 1: (c, 5)
• Point 2: (c, -2)

both of these points lie on a vertical line because they both have the x-coordinate of "c".

Vertical lines are distinctive not only in how they look but also in how they behave mathematically, especially when discussing slopes, which leads directly into the next concept.

Undefined Slope

The concept of slope is often described as the "rise over run". Mathematically, it is calculated using the formula \[m = \frac{y_2 - y_1}{x_2 - x_1}\]However, when dealing with vertical lines, there is a peculiarity that becomes evident.

• Since the x-coordinates of points on a vertical line are the same, \(x_2 - x_1\) equals zero.
• This leads to a denominator of zero in the slope calculation.

Hence, the slope of a vertical line is undefined because it involves division by zero. This is an important distinction in mathematics because an undefined slope signifies a vertical line.

Division by Zero

Division by zero is an operation that cannot be performed in mathematics. When you attempt to divide a number by zero, it results in an undefined mathematical expression or quantity. Consider this:

In the context of finding the slope of a line, if the calculation results in dividing a number by zero, such as:\[ m = \frac{-7}{0} \]it indicates a situation that is mathematically impossible.

• This arises when you have vertical lines, where the difference in x-values (\(x_2 - x_1\)) is zero.

The operation's impossibility gives the line's slope its undefined nature, reflecting the vertical orientation as discussed earlier. Understanding this concept helps you identify vertical lines and distinguish them from other types of slopes.