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Chapter 1: Problem 8

Find a parametric representation of the solution set of the linear equation. $$3 x-\frac{1}{2} y=9$$

Short Answer

Expert verified
The parametric representation of the equation is \(x = t\), \(y = 18 - 6t\).

Step by step solution

01

Isolate One Variable

In this case, the first step is to isolate \(y\). This can be done by moving \(3x\) to the right hand side of the equation to obtain: \(\frac{1}{2}y = 9 - 3x\). To get rid of the fraction, multiply through by 2: \(y = 18 - 6x\). This is called the 'Cartesian' form of the line because both variables are on one side of the equation.
02

Introduce Parametric Variable

As a next step, a parameter \(t\) is introduced to adjust the equation into parametric form. Simplest is to set \(t = x\). So, \(x = t\). Substitute this value into the equation for \(y\), and get \(y = 18 - 6t\).
03

Write Final Parametric Form

The final parametric form of the equation is written as a set of equations: \(x = t\) and \(y = 18 - 6t\). This is the parametric representation of the given linear equation.

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