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Chapter 1: Problem 9

Find a parametric representation of the solution set of the linear equation. $$x+y+z=1$$

Short Answer

Expert verified
The parametric representation of the solution set of the linear equation \(x + y + z = 1\) is \(x = t\), \(y = s\) and \(z = 1 - t - s\).

Step by step solution

01

Express the Equation as a System of Equations

The given equation is a simple linear equation in the three variables \(x\), \(y\), and \(z\). By expressing one of the variables in terms of the other two, a system of equations that can be solved parametrically can be created. Without losing generality, let's express \(z\) in terms of \(x\) and \(y\). Therefore, \(z = 1 - x - y\). The choice of \(z\) is arbitrary; you could use any of the three variables.
02

Identify and Assign Parameters

The next step is to identify parameters that can be used to express the solutions of the equation. For a single equation in three variables, we can introduce two parameters. Let鈥檚 use \(t\) and \(s\) as our parameters and set \(x = t\) and \(y = s\).
03

Express the Solution in Parametric Form

The final step is to express the solutions in terms of the parameters. Using the expressions from the previous steps, the solution is \(x = t\), \(y = s\) and \(z = 1 - t - s\). This is the parametric representation of the solution set.

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Most popular questions from this chapter

The augmented matrix represents a system of linear equations that has been reduced using Gauss-Jordan elimination. Write a system of equations with nonzero coefficients that is represented by the reduced matrix. $$\left[\begin{array}{lllr} 1 & 0 & 3 & -2 \\ 0 & 1 & 4 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]$$ There are many correct answers.

Use a computer software program or graphing utility to solve the system of linear equations. $$\begin{aligned} x_{1}-x_{2}+2 x_{3}+2 x_{4}+6 x_{5} &=6 \\ 3 x_{1}-2 x_{2}+4 x_{3}+4 x_{4}+12 x_{5} &=14 \\ x_{2}-x_{3}-x_{4}-3 x_{5} &=-3 \\ 2 x_{1}-2 x_{2}+4 x_{3}+5 x_{4}+15 x_{5} &=10 \\ 2 x_{1}-2 x_{2}+4 x_{3}+4 x_{4}+13 x_{5} &=13 \end{aligned}$$

Solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$\begin{array}{rr} -3 x+5 y= & -22 \\ 3 x+4 y= & 4 \\ 4 x-8 y= & 32 \end{array}$$

Determine the value(s) of \(k\) such that the system of linear equations has the indicated number of solutions. An infinite number of solutions \(4 x+k y=6\) \(k x+y=-3\)

(a) determine the polynomial function whose graph passes through the given points, and (b) sketch the graph of the polynomial function, showing the given points. $$(2,5),(3,2),(4,5)$$
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