Problem 38 - 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 \((3,1)\) and parallel to the vertical line passing through \((-1,-2)\)

Step-by-Step Solution

Answer

The equation is \(x = 3\), which is the same as \(1x + 0y = 3\) in standard form.

Step 1: Identify the Nature of the Line

Since the given line is parallel to another vertical line, the line in question must also be vertical. Vertical lines have an undefined slope.

Step 2: Determine the Equation of the Vertical Line

Vertical lines have the equation in the form of \(x = c\), where \(c\) is the x-coordinate of any point the line passes through. In this case, the line passes through point \((3, 1)\), so \(x = 3\).

Step 3: Put the Equation in Standard Form

The standard form of a line is \(Ax + By = C\). Since the equation of this line in step 2 is \(x = 3\), it can also be written as \(1x + 0y = 3\) to fit the standard form.

Key Concepts

Standard Form

The standard form of a line is a way to represent a linear equation in a specific structure. It's written as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers. This form is preferred for its neatness and simplicity in presenting the equation.
To convert any line equation to the standard form:

• First, move all terms involving variables to one side of the equation.
• Ensure that \( A \), the coefficient of \( x \), is a positive integer.
• Make sure \( A \), \( B \), and \( C \) are integers by clearing any fractions or decimals.

In the exercise, because the line parallel to a vertical line has the equation \( x = 3 \), it was written in standard form as \( 1x + 0y = 3 \). Here, \( A = 1 \), \( B = 0 \), and \( C = 3 \). Although it might not look like a typical linear equation with both \( x \) and \( y \) present, it still follows the rules of standard form. Remember, standard form is about structure, not the presence of both variables.

Parallel Lines

Parallel lines are lines in a plane that never meet. They're always the same distance apart. The defining feature of parallel lines is that they have identical slopes.
In Cartesian geometry, when two lines are parallel:

• If neither line is vertical, their slopes \( m \) will be equal. For example, if one line has the equation \( y = mx + b \), a parallel line will have the equation \( y = mx + c \), where \( c eq b \).
• If both lines are vertical, neither has a defined slope. But they share another feature: the equation \( x = c \) indicates they're always vertical.

In this exercise, the given line is parallel to a vertical line defined by \( x = -1 \). Therefore, it too must be vertical, sharing the distinct characteristic of vertical lines: an equation of the form \( x = c \). This guarantees they "run parallel" to each other in the sense that both follow the same vertical direction.

Vertical Lines

Vertical lines are a unique case of linear equations in coordinates. They run straight up and down, parallel to the y-axis. This is why, in geometry, they are considered special.
A vertical line can be easily recognized by its equation, which takes the form \( x = c \). Here, \( c \) is a constant and represents the x-coordinate that every point on the line shares.

• Because vertical lines do not "run" horizontally (no change in y), they have an undefined slope.
• This undefined slope is due to division by zero in slope formula calculations, as there's no horizontal movement.

In practical terms, for either math problems or real-world applications, if you're dealing with vertical lines, remember they break from typical rules of slope calculations.
In our example, because the line is passing through \( (3, 1) \) and is parallel to the vertical line at \( x = -1 \), this immediately dictated that the equation must be \( x = 3 \). Vertical lines keep their computations simple and substantially predictable.