Problem 39 - Calculus for Biology and Medicine

Question

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((1,-3)\) and perpendicular to the horizontal line passing through \((-1,-1)\)

Step-by-Step Solution

Answer

The standard form of the equation is \(x = 1\).

Step 1: Understand Perpendicular Line

A line that is perpendicular to a horizontal line is a vertical line. This means that the line we are looking for will have the same x-coordinate for all points on the line.

Step 2: Find x-coordinate of the Vertical Line

Since the line is vertical, all points on the line will have the same x-coordinate as the point it passes through. Given that the line passes through the point \((1, -3)\), the x-coordinate of the vertical line is 1.

Step 3: Write Equation of the Vertical Line

The equation of a vertical line can be written as \(x = a\), where \(a\) is the constant x-coordinate of all points on the line. So, the equation of the line is \(x = 1\).

Step 4: Convert to Standard Form

Standard form for the equation of a line is \(Ax + By = C\). For a vertical line \(x = 1\), it can be rewritten in standard form as \(1x + 0y = 1\), or simply \(x = 1\).

Key Concepts

Perpendicular Lines

Perpendicular lines are lines that intersect at a right angle, which is precisely 90 degrees. When dealing with perpendicular lines, one of the most important concepts is understanding how the slopes relate. To determine if two lines are perpendicular, you can check their slopes (often denoted as \(m\)). For two lines to be perpendicular, the product of their slopes must equal -1. This means:

• If the slope of one line is \(m\)
• The slope of the line perpendicular to it must be \(-\frac{1}{m}\)

Interestingly, horizontal and vertical lines always form a 90-degree angle when they meet, which makes them automatically perpendicular. While horizontal lines have a slope of 0, vertical lines have an undefined slope. This reinforces that they intersect perpendicularly as their intersection always creates a corner, similar to the edges of a square.

Vertical Lines

A vertical line is a straight line that extends in the up-and-down direction. In a vertical line, all points share the same \(x\)-coordinate, making it unique compared to other lines. If you imagine walking along a vertical line, you wouldn’t move left or right, only up or down. The equation representing a vertical line is \(x = a\). This equation tells you that for every point on this line, the \(x\) value remains the same and equals \(a\).

• If a vertical line passes through the point (3, 5), the equation is \(x = 3\).
• If it passes through (-1, -1), the equation is \(x = -1\).

These lines stand as a contrast to horizontal lines, where all points share the same \(y\)-coordinate, and these lines are neither slanted nor angled, making them very straightforward in equations.

Standard Form of a Line

Putting an equation in the standard form is a common way to express linear equations. The standard form looks like \(Ax + By = C\), where \(A\), \(B\), and \(C\) are real numbers.

• Here, \(A\) and \(B\) are coefficients, and \(C\) is the constant term.
• The equation is arranged so that \(x\) and \(y\) terms reside on the left and the constant on the right.

For vertical lines specifically, the standard form takes a slightly different look because there is no \(y\)-term. Take \(x = 1\) as an example. To express this as a standard form equation, it becomes \(1x + 0y = 1\), simplifying back to \(x = 1\). This shows how versatile the standard form is, as it can accommodate subtly different types of lines, preserving the fundamental aspects of linear equations.