Problem 101 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write an equation of the line passing through the two points. Use the slope- intercept form, if possible. If not possible, explain why. $$(4,-2),(4,5)$$

Step-by-Step Solution

Answer

The equation of the line passing through the points (4,-2) and (4,5) is \(x = 4\).

Step 1: Calculate the Slope

The slope of a line passing through two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by the formula \[m = (y_2-y_1)/(x_2-x_1)\]. Substituting the given points into this formula, we get \[m = (5-(-2))/(4-4)\].

Step 2: Analyze the Result from Step 1

The denominator in the slope formula is zero, which means the slope is undefined. This situation occurs when you're dealing with a vertical line.

Step 3: Write the Equation of the Line

Since the given points constitute a vertical line, the equation of the line will be of the form \(x = a\), where \(a\) is the x-coordinate of any point on the line. Thus, the equation of the line passing through the points (4,-2) and (4,5) is \(x = 4\). Note that it is not possible to write the equation in slope-intercept form \(y = mx + b\), because a vertical line has no slope.

Key Concepts

Slope

The slope, often represented as \(m\), is a measure of how steep a line is on a graph. It tells us how much \(y\) changes for a given change in \(x\). When examining two points, the formula to calculate the slope is \[m = \frac{y_2-y_1}{x_2-x_1}.\]Here:

• \(y_2\) and \(y_1\) are the y-coordinates of the points.
• \(x_2\) and \(x_1\) are the x-coordinates of the points.

This calculation gives a numerical value representing the line's direction:

• If \(m > 0\), the line ascends as you move from left to right.
• If \(m < 0\), the line descends as you move from left to right.
• If \(m = 0\), the line is horizontal.

However, when we encounter a division by zero, as in this exercise, it indicates a vertical line, resulting in an undefined slope.

Vertical Line

A vertical line is a unique type of straight line characterized by an undefined slope. Vertical lines occur when all points on the line share the same x-coordinate. Consider the two points (4, -2) and (4, 5) given in this exercise.
By substituting these points into our slope formula, you immediately notice that the denominator equals zero:\[m = \frac{5 - (-2)}{4 - 4} = \frac{7}{0},\]indicating a division by zero, and hence, an undefined slope.
The equation for a vertical line will always be of the form:

• \(x = a\)

where \(a\) is the shared x-coordinate. For our specific example, this equation is \(x = 4\). Vertical lines run parallel to the y-axis and do not sway to the left or right.
They cannot be expressed using the typical slope-intercept form.

Slope-Intercept Form

The slope-intercept form is a helpful way to express linear equations, especially when dealing with non-vertical lines. In its standard form, it's written as:\[y = mx + b,\]where:

• \(m\) is the slope.
• \(b\) is the y-intercept, representing where the line intersects the y-axis.

• The benefit of this form lies in its simplicity: a glance tells you the slope and where the line crosses the y-axis.

However, it's crucial to understand that this form cannot apply to vertical lines. Since vertical lines do not have a defined slope, attempting to fit them into this form results in frustration and incorrect equations.
In this task, because we determined our line is vertical, the slope-intercept form is unsuitable. Here, only the standard vertical line equation \(x = 4\) effectively represents the line.