Problem 54 - Calculus for Biology and Medicine

Question

Parameterize the equation of the line given in standard form. $$2 x-y+4=0$$

Step-by-Step Solution

Answer

The parametric equations are \(x = t\) and \(y = 2t + 4\).

Step 1: Identify the Standard Form

The equation given is in standard form: \(2x - y + 4 = 0\). This format is generally expressed as \(Ax + By + C = 0\), where \(A = 2\), \(B = -1\), and \(C = 4\).

Step 2: Solve for y-intercept

Rewrite the equation to solve for \(y\): \(-y = -2x - 4\) which simplifies to \(y = 2x + 4\). Now, we have the equation in slope-intercept form \(y = mx + b\), where \(m = 2\) and \(b = 4\).

Step 3: Parameterize the Line

Parameterize the line using a parameter \(t\). Choose a point where \(x = t\), and find \(y\) using the equation \(y = 2t + 4\). This gives the parametric equations as: \(x = t\) and \(y = 2t + 4\).

Step 4: Write the Parametric Equations

Converting the line to parametric form, we express it as: \(x(t) = t\) and \(y(t) = 2t + 4\), where \(t\) is a real number. These represent all the points \((x, y)\) on the line.

Key Concepts

Standard Form of a Line

The standard form of a line is a common way to express linear equations. It is written as \(Ax + By + C = 0\), where \(A\), \(B\), and \(C\) are constants, and \(A\) and \(B\) are not both zero. This format is especially useful for easily identifying lines in a two-dimensional plane.
To identify the standard form, you must recognize the components:

• \(A\): The coefficient of \(x\)
• \(B\): The coefficient of \(y\)
• \(C\): The constant

For example, with the equation \(2x - y + 4 = 0\), \(A = 2\), \(B = -1\), and \(C = 4\). Understanding this form helps in further manipulation, such as converting into the slope-intercept form or parametric equations.

Slope-Intercept Form

The slope-intercept form of a line is another popular way to express linear equations. It is especially useful for quickly understanding the slope and intercept of a line.
Slope-intercept form is expressed as:

• \(y = mx + b\)

Here, \(m\) represents the **slope** of the line. The slope indicates the rate at which \(y\) changes with respect to \(x\) and shows the line's steepness. For instance, with the equation \(y = 2x + 4\), the slope \(m\) is \(2\).
\(b\) is the **y-intercept**, which is the point where the line crosses the y-axis. In the example \(y = 2x + 4\), \(b\) is \(4\), meaning the line crosses the y-axis at the point \( (0, 4) \).
Transforming a standard form equation into slope-intercept form involves isolating \(y\) on one side of the equation, as shown in the step-by-step solution.

Parametric Equations

Parametric equations provide a framework for representing a line in terms of a parameter, often designated as \(t\). This method expresses both \(x\) and \(y\) coordinates as separate functions of \(t\), allowing a different perspective and flexibility in describing a line.
In our exercise, the line originally given in standard form \(2x - y + 4 = 0\) was parameterized by choosing \(x = t\). Then, the corresponding \(y\)-coordinate was determined using the equation derived in slope-intercept form:

• \(y = 2t + 4\)

This resulted in parametric equations:
\(x(t) = t\)
\(y(t) = 2t + 4\)
Where \(t\) can be any real number. These equations comprehensively describe the same line, providing an elegant way to capture the motion along the path of the line over a continuous variable \(t\). Parametric equations are widely used in areas such as physics and computer graphics, offering versatile ways to represent geometric concepts.