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Chapter 8: Problem 13

Eliminate the parameter \(t\) to rewrite the parametric equation as a Cartesian equation. $$ \left\\{\begin{array}{l} x(t)=5-t \\ y(t)=8-2 t \end{array}\right. $$

Short Answer

Expert verified
The Cartesian equation is \( y = 2x - 2 \).

Step by step solution

01

Solve for the Parameter t in terms of x

The given parametric equations are \( x = 5 - t \) and \( y = 8 - 2t \). First, solve the equation \( x = 5 - t \) for \( t \). Rearrange the equation to obtain \( t = 5 - x \).
02

Substitute for t in the Second Equation

Now that we have \( t = 5 - x \), substitute \( t \) into the second equation \( y = 8 - 2t \). Replace \( t \) with \( 5 - x \), which gives \( y = 8 - 2(5 - x) \).
03

Simplify the Expression

Simplify the expression from the substitution in the previous step. Distribute the \( -2 \) across the terms inside the parentheses: \( y = 8 - 10 + 2x \).
04

Simplify to Final Cartesian Equation

Combine like terms to find the final Cartesian equation. Simplify \( y = 8 - 10 + 2x \) to obtain \( y = 2x - 2 \). This is the Cartesian equation without parameter \( t \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cartesian equation
In mathematics, a Cartesian equation represents the relation between the coordinates of points on a graph. Cartesian equations are usually written in terms of the "x" and "y" variables that define a two-dimensional plane.
They are fundamental in graphing and analyzing linear or higher-degree equations.
For example, in the exercise, the Cartesian equation derived is \( y = 2x - 2 \).
This form helps in visualizing the line on a Cartesian plane, making it easier for students to understand the relationship between the variables "x" and "y".
eliminate parameter
Eliminating the parameter is a key technique used when working with parametric equations. Parametric equations define the coordinates of the points on a curve as functions of a parameter, often denoted as \( t \).
The goal is to express one variable solely as a function of the other variable, removing the parameter from the equations entirely.In our example, we start with two parametric equations:
  • \( x = 5 - t \)
  • \( y = 8 - 2t \)
To eliminate the parameter \( t \), we solve one of the equations for \( t \) and substitute it into the other. After doing so, we transform these equations into a Cartesian form without \( t \).
solve for variable
Solving for a variable involves isolating it on one side of the equation. This step is crucial when eliminating parameters from parametric equations to achieve a Cartesian form.
In the exercise, we start by solving \( x = 5 - t \) for \( t \).
By rearranging the equation, we get \( t = 5 - x \). This operation is the stepping stone to our next move: plugging this expression into the second equation \( y = 8 - 2t \).After substituting, we simplify the expression and get \( y = 2x - 2 \), which is now free from the parameter \( t \) and in a Cartesian form.
This Cartesian equation clearly shows the relationship between the variable \( x \) and the variable \( y \).

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Most popular questions from this chapter

As part of a video game, the point (7,3) is rotated counterclockwise about the origin through an angle of 40 degrees. Find the new coordinates of this point.

Find the angle between the vectors. $$ \langle 4,2\rangle ;\langle 8,4\rangle $$

Find the dot product of each pair of vectors. $$ \langle 2,-5\rangle ;\langle 8,-4\rangle $$

Parameterize the line from (4,1) to (6,-2) so that the line is at (4,1) at \(t=0,\) and at (6,-2) at \(t=1\)

Convert the given polar equation to a Cartesian equation. $$ r=3 \csc (\theta) $$
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