Problem 29 - 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 \((-1,-1)\) and parallel to the line passing through \((0,1)\) and \((3,0)\)

Step-by-Step Solution

Answer

The equation of the line is \(x + 3y = -4\).

Step 1: Find the Slope of the Given Line

The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For the line passing through \((0, 1)\) and \((3, 0)\), we find \(m = \frac{0 - 1}{3 - 0} = \frac{-1}{3}\).

Step 2: Use Slope to Write Equation of Parallel Line

Lines that are parallel have the same slope. Therefore, the line passing through \((-1, -1)\) and parallel to the given line will have a slope of \(-\frac{1}{3}\). Use the point-slope form of the line equation: \(y - y_1 = m(x - x_1)\), where \((x_1, y_1) = (-1, -1)\). This gives: \(y - (-1) = -\frac{1}{3}(x - (-1))\).

Step 3: Simplify the Equation

Simplify the equation from Step 2. Start by distributing: \(y + 1 = -\frac{1}{3}(x + 1)\) to get \(y + 1 = -\frac{1}{3}x - \frac{1}{3}\). Then, isolate \(y\) to get \(y = -\frac{1}{3}x - \frac{4}{3}\).

Step 4: Convert to Standard Form

The standard form of a line is \(Ax + By = C\), with integer coefficients, and \(A\), \(B\), and \(C\) having no common factors other than 1. Start by eliminating fractions by multiplying through by 3: \(3y = -x - 4\). Reorder to \(x + 3y = -4\). The equation \(x + 3y = -4\) is now in standard form.

Key Concepts

Understanding Slope

The concept of slope is fundamental to understanding linear equations. Slope tells us how steep a line is and whether it will rise or fall as you move from left to right.
When you find the slope of a line connecting two points, \((x_1, y_1)\) and \((x_2, y_2)\), you use the formula:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

The slope is often represented by the letter \(m\).
If \(m\) is positive, the line ascends as you move along the x-axis. If \(m\) is negative, the line descends.

Parallel Lines and Slope

Lines that are parallel share the same slope.
In this context, when you want a line parallel to another, ensure it has the same slope. For instance, if a given line has \(m = -\frac{1}{3}\), any line parallel to it will also have \(m = -\frac{1}{3}\).
This is crucial for maintaining the distance between the lines over all x-values.

Exploring the Point-Slope Form

The point-slope form is a convenient way to write the equation of a line when you have one point and the slope.
The formula is generally given as:

• \(y - y_1 = m(x - x_1)\),

where \((x_1, y_1)\) is a known point on the line and \(m\) is the slope.
This form is especially useful for developing equations of lines that are parallel or perpendicular to a given line.

Applying Point-Slope Form

To illustrate its use, consider a line with a known point \((-1, -1)\) and a slope of \(-\frac{1}{3}\).
Applying point-slope form, the equation would be:

• \(y + 1 = -\frac{1}{3}(x + 1)\)

This equation is straightforward and only requires basic multiplication and addition to solve, making it a handy tool in many problems.

The Standard Form of a Line

When discussing the equation of a line, the standard form is another common way to write it.
This form expresses the equation as:

• \(Ax + By = C\),

where \(A\), \(B\), and \(C\) are integers, and \(A\) is usually positive.
One of the benefits of this form is its ease of turning equations into integers, which aids readability and calculation in some contexts.

Transforming to Standard Form

To convert an equation from the point-slope form or slope-intercept form to standard form, you may need to multiply to clear fractions and rearrange terms.
For example, from \(y = -\frac{1}{3}x - \frac{4}{3}\), multiply every term by 3 to remove fractions resulting in \(3y = -x - 4\).
Rearrange it to \(x + 3y = -4\) ensuring all coefficients are integers and \(x\) is positive.