Problem 12 - 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 \((-1,4)\) and \((1,-4)\)

Step-by-Step Solution

Answer

The equation in standard form is \(4x + y = 0\).

Step 1: Calculate the Slope

The first step is to find the slope of the line passing through the points \((-1,4)\) and \((1,-4)\). The formula for the slope \(m\) is \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. Substitute \(x_1 = -1\), \(y_1 = 4\), \(x_2 = 1\), and \(y_2 = -4\) into the formula: \[ m = \frac{-4 - 4}{1 - (-1)} = \frac{-8}{2} = -4 \]. So, the slope of the line is \(-4\).

Step 2: Use the Point-Slope Form

With the slope \(m = -4\) and one of the points \((-1, 4)\), use the point-slope form of the equation of a line, given by \( y - y_1 = m(x - x_1) \). Substitute the known values into the formula: \[ y - 4 = -4(x + 1) \].

Step 3: Simplify to Slope-Intercept Form

Expand the equation from Step 2: \( y - 4 = -4x - 4 \). Simplify by adding 4 to both sides to obtain \( y = -4x \).

Step 4: Convert to Standard Form

The standard form of a linear equation is \( Ax + By = C \). Rearrange \( y = -4x \) to match this form: \( 4x + y = 0 \). This is the equation of the line in standard form.

Key Concepts

Slope of a Line

Understanding the slope of a line is crucial in determining how steep a line is. The slope, represented by the letter \( m \), measures the change in the vertical direction (the rise) compared to the change in the horizontal direction (the run). In simpler terms, it tells us how much a line goes up or down as we move from left to right.

To calculate the slope of a line that passes through two points, \((x_1, y_1)\) and \((x_2, y_2)\), we use the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This formula helps us find out exactly how much the line rises or falls between these two points.
In the given exercise, the points are \( (-1, 4) \) and \( (1, -4) \), and plugging these values into the formula gives us a slope of \(-4\). This means that for every step we take to the right, the line goes down 4 steps.

Point-Slope Form

Once we know the slope, we can write an equation for the line using the point-slope form. This form is particularly useful because it directly incorporates the slope of the line \( m \) and a specific point \((x_1, y_1)\) through which the line passes. The point-slope form of a line is expressed as:

• \( y - y_1 = m(x - x_1) \)

Insert the slope \( m = -4 \) and one of the points, say \((-1, 4)\), into the equation:
\( y - 4 = -4(x + 1) \).

This equation captures both the steepness of the line and its position. It is a great starting step when you have a point and a slope for your line.

Slope-Intercept Form

To further simplify the line equation, we can convert the point-slope form into the slope-intercept form. This form is very popular because it easily shows both the slope of the line and its intersection with the y-axis. The general form of the slope-intercept equation is:

• \( y = mx + b \)

Here, \( m \) still represents the slope, and \( b \) is the y-intercept (the point where the line crosses the y-axis).

From our point-slope form discussed earlier:
When simplified, \( y - 4 = -4(x + 1) \) expands and rearranges to give \( y = -4x\). In this case, \( b \) is 0, indicating that the line passes through the origin.

Standard Form

The standard form of a line's equation is often used in more formal or algebraic contexts. Unlike the slope-intercept form, which emphasizes clarity through the slope and y-intercept, the standard form displays the equation as a balance between the x and y terms. Standard form is typically written as:

• \( Ax + By = C \)

Where \( A \), \( B \), and \( C \) are integers, and \( A \) should be non-negative.

To convert the simpler slope-intercept form \( y = -4x \) into standard form:
Rearrange the terms to make zero on one side, resulting in \( 4x + y = 0 \).
This transformation shows the line equation suits the structured nature of the standard form, aiding in instances such as graph interpretation or further algebraic manipulation.