In Exercises 1–8, parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t.x = 3 − 5t, y = 4 + 2t; t = 1

Verified step by step guidance

1. Identify the given parametric equations: \(x = 3 - 5t\) and \(y = 4 + 2t\), and the given parameter value \(t = 1\).
2. Substitute the value of \(t = 1\) into the equation for \(x\): calculate \(x = 3 - 5(1)\).
3. Substitute the value of \(t = 1\) into the equation for \(y\): calculate \(y = 4 + 2(1)\).
4. Simplify both expressions to find the numerical values of \(x\) and \(y\) at \(t = 1\).
5. Write the coordinates of the point on the curve as \((x, y)\) using the values found in the previous step.

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.

Substitution of Parameter Values

To find a specific point on a parametric curve, substitute the given parameter value into the parametric equations. This yields the corresponding x and y coordinates, pinpointing the exact location on the curve for that parameter.

Coordinate Geometry of Plane Curves

Understanding how coordinates (x, y) represent points on a plane is essential. By evaluating parametric equations at a given t, you determine a point's position in the Cartesian plane, which helps visualize and analyze the curve's shape and behavior.