Problem 54 - Calculus for Biology and Medicine

Question

Parameterize the equation of the line given in standard form. $$ x-5 y+7=0 $$

Step-by-Step Solution

Answer

The parametric equations are \(x = t\) and \(y = \frac{1}{5}t + \frac{7}{5}\).

Step 1: Understand the Standard Form

The standard form of a linear equation in two variables is given as \(Ax + By + C = 0\). In the given equation, \(A=1\), \(B=-5\), and \(C=7\).

Step 2: Solve for y

To parameterize, we first need to express \(y\) in terms of \(x\). Solve the equation \(x - 5y + 7 = 0\) for \(y\). Move \(x\) and \(7\) to the other side: \(-5y = -x - 7\). Now divide every term by \(-5\), resulting in \(y = \frac{1}{5}x + \frac{7}{5}\).

Step 3: Parameterize the Equation using a Parameter t

We introduce a parameter \(t\) to express both \(x\) and \(y\) in terms of \(t\). Set \(x = t\). Using the expression for \(y\) from Step 2, substitute \(x\) with \(t\): \(y = \frac{1}{5}t + \frac{7}{5}\).

Step 4: Write the Parametric Equations

The parameterized equations are: \(x = t\) and \(y = \frac{1}{5}t + \frac{7}{5}\), where \(t\) is any real number.

Key Concepts

Linear Equations and Their Forms

Linear equations form the foundation of algebra, representing relationships where each variable occurs to the first power, and their graph is a straight line. The most common forms of a linear equation are:

• Standard Form: This is written as \(Ax + By + C = 0\). Here, \(A\), \(B\), and \(C\) are constants, and \(x\), \(y\) are the variables.
• Slope-Intercept Form: This is expressed as \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.

In any form, a linear equation describes a constant rate of change, which is depicted as a straight line on a graph. Understanding each form allows us to solve, convert, and analyze the equation for different applications.

Parametric Equations Simplified

Parametric equations provide a flexible way to express the equations of lines, focusing on the role of a parameter. Unlike the standard or slope-intercept forms, parametric equations introduce a third variable, usually \(t\), to define the x and y coordinates. This is particularly useful when:

• Representing motion: Where each variable depends on time \(t\).
• Simplifying complex curves or transforming equations for easier computation.

To parameterize a line, two equations are established such that \(x = f(t)\) and \(y = g(t)\), effectively linking the line's position to the parameter \(t\). It translates the problem of finding points on the line into finding the appropriate \(t\) values, enhancing the comprehension of geometric shapes.

Coordinate Geometry and Lines

Coordinate geometry, or analytic geometry, utilizes an ordered pair of numbers (coordinates) to pinpoint the location of points on a plane. It elegantly merges algebra and geometry, allowing for the exploration of geometric shapes through algebraic equations. Through coordinate geometry, we analyze:

• Points: Defined by coordinates \((x, y)\).
• Lines: Expressed algebraically, such as in parametric form, which makes understanding shifts, stretches, and rotations intuitive.
• Distance and Midpoints: Calculated using coordinate principles to determine lengths and origins swiftly.

This methodology revolutionizes our understanding of shapes and their properties, providing a visual representation that is considerably more insightful than numerical calculations alone. It gives a graphical consistency to real-world tasks, from design to physics and beyond.