Problem 22 - 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 \(-1 / 3\) and \(y\) -intercept \((0,1 / 3)\)

Step-by-Step Solution

Answer

The standard form of the line is \( x + 3y = 1 \).

Step 1: Understand the Slope-Intercept Equation

The slope-intercept form of a line's equation is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. For this exercise, we have a slope \( m = -\frac{1}{3} \) and a y-intercept \( b = \frac{1}{3} \).

Step 2: Plug the Information into the Slope-Intercept Formula

Substitute \( m = -\frac{1}{3} \) and \( b = \frac{1}{3} \) into the equation \( y = mx + b \) to get \( y = -\frac{1}{3}x + \frac{1}{3} \).

Step 3: Convert the Equation to Standard Form

The standard form of a line is \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) should be non-negative. First, eliminate the fraction by multiplying the entire equation by 3: \( 3y = -x + 1 \).

Step 4: Rearrange to Obtain the Standard Form

Rearrange \( 3y = -x + 1 \) to express it in the order \( Ax + By = C \). Add \( x \) to both sides to get \( x + 3y = 1 \), which is the equation in standard form.

Key Concepts

Slope-Intercept Form

One of the most common ways to express the equation of a line is through the slope-intercept form. This form is written as \( y = mx + b \). Here, \( m \) represents the slope of the line, which indicates the line’s steepness and direction.
The parameter \( b \) is the y-intercept, the point where the line crosses the y-axis. Understanding this form is essential because it quickly shows both the rate of change and the starting point of a line.

• The slope \( m \) can describe a positive or negative inclination, with positive slopes rising from left to right, and negative ones falling.
• The y-intercept \( b \) tells where the line meets the y-axis, which happens when \( x = 0 \).

In our exercise, the line has a slope of \(-\frac{1}{3}\), meaning it gently descends as it moves to the right, and a y-intercept of \(\frac{1}{3}\), which reveals it crosses the y-axis at this point.

Standard Form of a Line

Sometimes, a line's equation is more conveniently expressed in the standard form, \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers. This form is particularly useful for quickly finding intersections and performing operations with other lines.
For an equation to be in standard form:

• A should be a non-negative integer.
• A, B, and C should ideally have no common factors besides 1.

In the given exercise, converting from the slope-intercept form \( y = -\frac{1}{3}x + \frac{1}{3} \) to standard form involves eliminating fractions. This is achieved by multiplying each term by a common multiple like 3. After rearranging, you arrive at the equation \( x + 3y = 1 \).
This form is neat and integer-only, making it perfect for various algebraic functions.

Linear Equations

Linear equations are fundamental in understanding patterns and relationships in algebra. They describe straight lines and can be expressed in multiple forms, including the slope-intercept form and the standard form, which we discussed. Linear equations have crucial properties:

• They graph as a straight line in a coordinate plane.
• The general form is \( y = mx + b \) or \( Ax + By = C \).
• They show a constant rate of change, represented by the slope.

Equations like \( y = -\frac{1}{3}x + \frac{1}{3} \) or \( x + 3y = 1 \) exemplify how we can translate real-world scenarios into mathematical descriptions. The solution to a linear equation is all the points \((x, y)\) that lie along the line. Understanding these concepts makes it easier to tackle various problems in algebra and other areas of math.