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Chapter 10: Problem 75

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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.

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