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=2 t,y=3t−4;−10≤t≤10 

Verified step by step guidance

1. Start with the given parametric equations: \(x = 2t\) and \(y = 3t - 4\) where \(-10 \leq t \leq 10\).
2. Solve the first equation for the parameter \(t\): from \(x = 2t\), we get \(t = \frac{x}{2}\).
3. Substitute \(t = \frac{x}{2}\) into the second equation \(y = 3t - 4\) to eliminate the parameter \(t\). This gives \(y = 3 \left( \frac{x}{2} \right) - 4\).
4. Simplify the expression to get the relationship between \(x\) and \(y\): \(y = \frac{3}{2}x - 4\).
5. Describe the curve: since the equation is linear in \(x\) and \(y\), the curve is a straight line with slope \(\frac{3}{2}\) and \(y\)-intercept \(-4\). The parameter range \(-10 \leq t \leq 10\) corresponds to \(x\) values between \(-20\) and \(20\).

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, usually denoted 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.

Curve Description from Equations

Describing the curve means interpreting the resulting equation or parametric form to identify its geometric shape, such as a line, circle, or parabola. Understanding the domain of t helps determine the portion of the curve represented.