Problem 21 - 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 \(1 / 2\) and \(y\) -intercept \((0,2)\)

Step-by-Step Solution

Answer

The standard form of the line is \(x - 2y = -4\).

Step 1: Identify the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In this problem, the slope \(m = \frac{1}{2}\), and the y-intercept \(b = 2\).

Step 2: Plug Values into Slope-Intercept Equation

Substitute the slope \(m = \frac{1}{2}\) and y-intercept \(b = 2\) into the slope-intercept equation. This gives us \(y = \frac{1}{2}x + 2\).

Step 3: Convert to Standard Form

The standard form of a line is \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) should be positive. Starting from \(y = \frac{1}{2}x + 2\), multiply through by 2 to eliminate the fraction: \(2y = x + 4\).

Step 4: Rearrange to Obtain Standard Form

Rearrange \(2y = x + 4\) by subtracting \(x\) from both sides to get \(-x + 2y = 4\). To make \(A\) positive, multiply the entire equation by -1. Thus, the standard form is \(x - 2y = -4\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most common ways to express the equation of a line. It's given by the equation \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept. This form is highly useful because it provides clear information about the line at a glance. You can readily see how steep the line is by looking at the slope \(m\), and where it crosses the y-axis by checking the y-intercept \(b\).
For example, in an equation like \(y = \frac{1}{2}x + 2\),\( m = \frac{1}{2}\), indicates that for each step to the right, the line moves half a step up. The \(b = 2\) shows the line crosses the y-axis at the point \((0, 2)\). This makes it very easy to both graph the line and understand its behavior.
When you want to quickly determine the slope or intercept of a line from its equation or graph, the slope-intercept form is your go-to choice. It's straightforward and gives immediate insight into the line's properties.

Standard Form of a Line

The standard form of a line is another way to write the equation of a line, given by \(Ax + By = C\). In this format, \(A\), \(B\), and \(C\) are integers, and it's customary for \(A\) to be a positive number.
This form can be especially beneficial for solving systems of equations or performing algebraic manipulations. Unlike the slope-intercept form, which shows slope and intercept explicitly, the standard form encapsulates the line's properties less transparently. However, it is quite versatile when working with analytical calculations.
As an example, transition from the slope-intercept form \(y = \frac{1}{2}x + 2\) to standard form, we rearrange it to the form \(x - 2y = -4\). Here, \(A = 1\), \(B = -2\), and \(C = -4\). This makes the line ready for further algebraic manipulation or when comparing different lines in standard form.

Y-Intercept

The y-intercept of a line is a crucial concept in understanding its position in the coordinate plane. The y-intercept is the point where the line crosses the y-axis. In other words, it’s the value of \(y\) when \(x=0\).
In the slope-intercept form \(y = mx + b\), \(b\) signifies the y-intercept. For the given equation \(y = \frac{1}{2}x + 2\), the y-intercept \((0, 2)\) tells us that the line will pass through the point \((0, 2)\) on the coordinate plane.
Understanding the y-intercept is important because it provides a starting point for drawing a line. It’s also part of many real-world applications, for instance, determining the starting value of a dependent variable before any independent changes take place. Recognizing the y-intercept quickly aids in both graphing lines and solving equations more effectively.