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

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Identify the given parametric equations: \(x = 3 - 5t\) and \(y = 4 + 2t\), and the given parameter value \(t = 1\).

Substitute the value of \(t = 1\) into the equation for \(x\): calculate \(x = 3 - 5(1)\).

Substitute the value of \(t = 1\) into the equation for \(y\): calculate \(y = 4 + 2(1)\).

Simplify both expressions to find the numerical values of \(x\) and \(y\) at \(t = 1\).

Write the coordinates of the point on the curve as \((x, y)\) using the values found in the previous step.

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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.