Final answer:
To eliminate the parameter in a set of equations, one often solves for the parameter in terms of one variable and substitutes it into the other equation, which can lead to a manageable Cartesian equation. This simplification process is crucial in various applications such as motion equations, line integrals, and circuit analysis.
Step-by-step explanation:
To eliminate the parameter and find a simplified Cartesian equation, we start with equations that express x and y in terms of a parameter, commonly 't'. The goal is to eliminate 't' to write y as a function of x or vice versa. When you have motion equations or line integrals, you often have a choice of which variable to eliminate based on the simplicity of resulting expressions.
For example, if you have the equations x = f(t) and y = g(t), you will first solve one of the equations for t, say t = h(x), and then substitute this into the other equation to get y as a function of x. If the given equations are already in the form of a quadratic, such as x² + 0.0211x - 0.0211 = 0, the quadratic formula or simplification techniques could be applied to solve for the variable x outright.
Similarly, when working with electrical circuits or transmission coefficients, you divide your equations by a constant to simplify them sufficiently to solve for the unknown quantities. Mathematics often involves this process of simplification and substitution to reach a more manageable form of an equation.