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

Step 1: Identify the Slope-Intercept Form

The equation of a line in slope-intercept form is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In this problem, we know the slope \( m = \frac{1}{5} \).

Step 2: Use the x-intercept to Find y-intercept

Given that the line passes through the x-intercept \( \left(-\frac{1}{2}, 0\right) \), substitute \( x = -\frac{1}{2} \) and \( y = 0 \) into the slope-intercept form to find the y-intercept \( b \): \[ 0 = \frac{1}{5}\left(-\frac{1}{2}\right) + b \]. This simplifies to \[ 0 = -\frac{1}{10} + b \]. Thus, \( b = \frac{1}{10} \).

Step 3: Write the Slope-Intercept Form

Now that we have \( m = \frac{1}{5} \) and \( b = \frac{1}{10} \), the slope-intercept form of the line is \( y = \frac{1}{5}x + \frac{1}{10} \).

Step 4: Convert to Standard Form

The standard form of a line's equation is \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers. Starting from \( y = \frac{1}{5}x + \frac{1}{10} \), multiply every term by 10 to eliminate fractions: \( 10y = 2x + 1 \). Reorder this to get \(-2x + 10y = 1 \). Multiply through by -1 to get positive \( A \): \[ 2x - 10y = -1 \].

Key Concepts

slope-intercept form

The slope-intercept form of a linear equation is one of the most commonly used formats to express linear equations. It is written as \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept. This form is incredibly useful because it tells us two important things right away:

• Slope \((m)\): It describes how steep the line is. The slope is calculated as the 'rise' over 'run,' or the change in \( y \) over the change in \( x \).
• Y-intercept \((b)\): It is the point where the line crosses the y-axis. In other words, it's the value of \( y \) when \( x = 0 \).

To use the slope-intercept form, you need to know either the slope and one point on the line or both the slope and the y-intercept. In our exercise, we have the slope \( m = \frac{1}{5} \) and use the x-intercept to find the y-intercept \( b = \frac{1}{10} \), giving us the equation \( y = \frac{1}{5}x + \frac{1}{10} \). This form makes it easy to graph the line as well on a coordinate plane.

x-intercept

In any linear equation, the x-intercept is the point where the line crosses the x-axis. This means that at the x-intercept, the value of \( y \) is zero. Knowing the x-intercept is incredibly useful because it provides a direct way to determine other aspects of the linear equation.

• Identifying the Intercept: The x-intercept is typically represented as \((x, 0)\). For instance, in this problem, the x-intercept is given as \( \left(-\frac{1}{2}, 0\right) \).
• Using the Intercept: You can substitute the x-intercept into the slope-intercept form \( y = mx + b \) to solve for the y-intercept \( b \). Using our example, substituting \( x = -\frac{1}{2} \) and \( y = 0 \) allows you to solve for \( b \), leading to \( b = \frac{1}{10} \).

This method of utilizing intercepts makes linear equations versatile and relatively straightforward to handle, especially when transitioning between different forms of equations.

standard form

The standard form of a linear equation is represented as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) is non-negative. This form is particularly useful for analysis and comparison between different linear equations.

• Conversion Process: To convert from the slope-intercept form \( y = mx + b \) to the standard form, you need to rearrange the equation. For instance, from \( y = \frac{1}{5}x + \frac{1}{10} \), you would first eliminate fractions by multiplying through by 10, resulting in \( 10y = 2x + 1 \).
• Reordering Terms: Next, you rearrange the terms to have \( Ax \) and \( By \) on one side: \( -2x + 10y = 1 \). Adjust so that the coefficient of \( x \) is positive: \( 2x - 10y = -1 \).
• Practical Benefits: Standard form is favored in certain applications because it makes it clear to see integer coefficients, which are easier to work with for some calculations, such as finding intercepts using specific methods.

The conversion to standard form thus rounds out our understanding of linear representations, offering another perspective on the relationships between variables.