11–14. Working with parametric equations Consider the following parametric equations.
c. Eliminate the parameter to obtain an equation in x and y.
d. Describe the curve.


x=−t+6, y=3t−3; −5≤t≤5 

Verified step by step guidance

1. Start with the given parametric equations: \(x = -t + 6\) and \(y = 3t - 3\) with the parameter \(t\) in the interval \([-5, 5]\).
2. Solve the first equation for \(t\): from \(x = -t + 6\), rearrange to get \(t = 6 - x\).
3. Substitute this expression for \(t\) into the second equation: \(y = 3(6 - x) - 3\).
4. Simplify the equation for \(y\) to express it solely in terms of \(x\): \(y = 18 - 3x - 3\), which simplifies further to \(y = 15 - 3x\).
5. Interpret the resulting equation \(y = 15 - 3x\): this is a linear equation representing a straight line. The parameter interval \(-5 \leq t \leq 5\) will restrict the portion of the line that corresponds to the curve.

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, usually denoted as t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves and motions.

Eliminating the Parameter

Eliminating the parameter involves manipulating the parametric equations to remove t, resulting in a direct relationship between x and y. This often requires solving one equation for t and substituting into the other, yielding a Cartesian equation of the curve.

Analyzing and Describing the Curve

Describing the curve means interpreting the resulting Cartesian equation or parametric form to identify its shape, type (line, circle, etc.), and domain restrictions. Understanding the parameter range helps determine the portion of the curve represented.