Problem 26 - 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 with slope \(1 / 5\) and \(x\) -intercept \((-1 / 2,0)\)

Step-by-Step Solution

Answer

The standard form of the line is \( 2x - 10y = -1 \).

Step 1: Understand the Requirements

We need to find the equation of a line with a given slope of \( \frac{1}{5} \) and an \( x \)-intercept at \( \left( -\frac{1}{2}, 0 \right) \). The line's equation should be expressed in standard form, \( Ax + By = C \).

Step 2: Use the Point-Slope Form

The point-slope form of a line's equation is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. Substituting \( m = \frac{1}{5} \) and \( (x_1, y_1) = \left( -\frac{1}{2}, 0 \right) \), we get:\[ y - 0 = \frac{1}{5}(x + \frac{1}{2}) \]

Step 3: Simplify the Equation

Solve for \( y \) by distributing and simplifying the right side:\[ y = \frac{1}{5}x + \frac{1}{5} \times \frac{1}{2} \]\[ y = \frac{1}{5}x + \frac{1}{10} \]

Step 4: Convert to Standard Form

The standard form is \( Ax + By = C \). We start by eliminating the fractions by multiplying the entire equation by 10:\[ 10y = 2x + 1 \]Rearrange it to get the standard form:\[ -2x + 10y = 1 \]

Step 5: Adjust Standard Form

To make \( A \) positive (a common convention), multiply the entire equation by -1:\[ 2x - 10y = -1 \]

Key Concepts

Point-Slope Form

The point-slope form is a great tool to find the equation of a line when you know one point on the line and the slope. Its general structure is given by \( y - y_1 = m(x - x_1) \), where:

• \( m \) is the slope of the line
• \((x_1, y_1)\) is a known point on the line

This form directly connects the slope and a specific point to the line's equation.
For example, suppose we have the slope of the line as \( \frac{1}{5} \) and a point \( (-\frac{1}{2}, 0) \).
By inserting these values into the point-slope formula, we get the equation \( y - 0 = \frac{1}{5} (x + \frac{1}{2}) \).
This is a direct application, allowing easy translation to other forms like the slope-intercept or the standard form.

Standard Form of a Line

The standard form of a line's equation is expressed as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) is preferably a positive integer.
This form is often used in computations and is ideal for quickly identifying intercepts.
Transforming other forms of equations into standard form usually involves clearing fractions and organizing terms. In our exercise, we start with:

• The slope-intercept equation \( y = \frac{1}{5}x + \frac{1}{10} \)
• Multiply by 10 to clear the fractions: \( 10y = 2x + 1 \)
• Reorganize terms to achieve \( -2x + 10y = 1 \)
• Finally, ensuring \( A \) is positive, multiply through by -1 to get \( 2x - 10y = -1 \)

This process emphasizes an important trait of the standard form: its robustness when handling integer coefficients, which can help simplify certain types of analyses.

Slope-Intercept Form

The slope-intercept form is one of the most commonly used forms for equations of lines because of its simplicity and directness.
The general structure is given by \( y = mx + b \), where:

• \( m \) represents the slope of the line
• \( b \) represents the y-intercept

This form is excellent for quickly identifying both the rate of change and where the line crosses the y-axis.
For a line with a slope of \( \frac{1}{5} \) and after processing through the point-slope form, one can solve and simplify to get \( y = \frac{1}{5}x + \frac{1}{10} \).
This makes it easy to visualize how the line rises over run \((\text{slope})\) and at which point it crosses the y-axis. It emphasizes the linear relationship between \( y \) and \( x \), providing a practical approach for graphing and interpreting line equations.