Problem 75 - Algebra and Trigonometry

Question

Explain how the rectangular equation \(y=5 x\) can have infinitely many sets of parametric equations.

Step-by-Step Solution

Answer

The rectangular equation \(y=5x\) can have infinitely many sets of parametric equations as we can introduce a parameter 't' to represent an infinite number of possible x and y pairings that satisfy the equation. For example, if \(x=t\), then \(y=5t\). If \(x=2t\), then \(y=10t\), and so on for infinitely many values of 't'.

Step 1: Understand the Relationship between Rectangular and Parametric Equations

Rectangular equations represent relationships between two variables, such as y and x. Parametric equations, on the other hand, express these variables in terms of a third variable, usually denoted as 't' which can take on infinite values.

Step 2: Parameterize the Equation

To write parametric equations for the line \(y = 5x\), we introduce a parameter 't'. We can then let \(x=t\) and \(y=5t\). These represent one possible set of parametric equations for the given rectangular equation.

Step 3: Show Infinite Parametric Equations

However, we are not restricted to letting \(x=t\). We could also let \(x=2t\), \(x=3t\), \(x=0.5t\), etc. and then \(y\) would be \(5\) times these. This shows there are infinitely many sets of parametric equations for the given rectangular equation.