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Chapter 1: Problem 7

Find a parametric representation of the solution set of the linear equation. $$2 x-4 y=0$$

Short Answer

Expert verified
The parametric equations representing the given linear equation are \(x = 2t\) and \(y = t\).

Step by step solution

01

Solve the equation for one variable

A common way to approach this is to solve the equation for one variable first. Here it is easier to solve for \(x\), giving us the equation \(x = 2y\).
02

Substitute

Next, one can substitute \(y = t\) (where \(t\) is the parameter) into this equation to find the corresponding value for \(x\). So, we get that \(x = 2t\).
03

Parametric equations

Finally, the parametric equations representing the line are \(x = 2t\) and \(y = t\). This represents all possible solutions to the linear equation.

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Most popular questions from this chapter

Solve the system of linear equations for \(x\) and \(y\). $$\begin{aligned} (\cos \theta) x+(\sin \theta) y &=1 \\ (-\sin \theta) x+(\cos \theta) y &=0 \end{aligned}$$

Use a computer software program or graphing utility to solve the system of linear equations. $$\begin{aligned} x_{1}+2 x_{2}-2 x_{3}+2 x_{4}-x_{5}+3 x_{6} &=0 \\ 2 x_{1}-x_{2}+3 x_{3}+x_{4}-3 x_{5}+2 x_{6} &=17 \\ x_{1}+3 x_{2}-2 x_{3}+x_{4}-2 x_{5}-3 x_{6} &=-5 \\ 3 x_{1}-2 x_{2}+x_{3}-x_{4}+3 x_{5}-2 x_{6} &=-1 \\ -x_{1}-2 x_{2}+x_{3}+2 x_{4}-2 x_{5}+3 x_{6} &=10 \\ x_{1}-3 x_{2}+x_{3}+3 x_{4}-2 x_{5}+x_{6} &=11 \end{aligned}$$

Is it possible for a system of linear equations with fewer equations than variables to have no solution? If so, give an example.

Find the solution set of the system of linear equations represented by the augmented matrix. $$\left[\begin{array}{rrrr} 1 & 2 & 1 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right]$$

(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. $$(2,4),(3,4),(4,4)$$
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