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 \( y = 4 + 2t \).
2. Substitute the given value of \( t = 1 \) into the equation for \( x \): \( x = 3 - 5(1) \).
3. Calculate the expression for \( x \) after substitution.
4. Substitute the given value of \( t = 1 \) into the equation for \( y \): \( y = 4 + 2(1) \).
5. Calculate the expression for \( y \) after substitution.

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a variable, often denoted as 't'. In this case, x and y are defined in terms of t, allowing for the representation of curves in a two-dimensional plane. Understanding how to manipulate these equations is essential for finding specific points on the curve.

Substitution

Substitution is a fundamental algebraic technique used to evaluate expressions by replacing a variable with a specific value. In the context of parametric equations, substituting the given value of t into the equations for x and y allows us to calculate the corresponding coordinates of the point on the curve.

Coordinate System

A coordinate system provides a framework for locating points in a plane using pairs of numbers (x, y). In this exercise, understanding the Cartesian coordinate system is crucial, as it allows us to interpret the results of the parametric equations and visualize the point's position in relation to the axes.