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The elimination method is a popular technique for solving systems of equations, both linear and nonlinear. It allows you to reduce the number of variables by combining equations in a way that cancels out a particular variable. This can simplify the problem significantly.

Here's how it works:

• Identify pairs of equations, focusing on terms that can be added or subtracted to eliminate a variable.
• Combine these equations using addition or subtraction to cancel one of the variables.
• In this example, adding the equations resulted in the elimination of the variable \( y^2 \), while subtracting eliminated \( x^2 \).

The primary goal of elimination is to transform a system into simpler, more manageable equations. By repeatedly applying elimination, you can reduce a system with multiple variables to one with a single variable, allowing for straightforward solutions.

A system of equations consists of multiple equations that share common variables. Solving a system typically involves finding values for the variables that satisfy every equation simultaneously.

In our example, we dealt with a nonlinear system defined by:

• \( 4x^2 - 9y^2 = 36 \)
• \( 4x^2 + 9y^2 = 36 \)

These are two separate equations with common variables \( x \) and \( y \). When solving systems, your goal is to determine variable values where both equations hold true. Successfully solving the system will give you a set of solutions, typically expressed as ordered pairs (or triples if there are three variables), showing all potential combinations of variables that satisfy the equations.

Systems can be linear or nonlinear, depending on whether the equations include terms higher than the first degree, such as squares (\(x^2\), \(y^2\)) or other non-linear expressions.

Quadratic equations are polynomials that take the form \(ax^2 + bx + c = 0\). They can be solved using various methods such as factoring, completing the square, quadratic formula, or graphing.

In our example, one of the steps involved distilling a quadratic-like form \(8x^2 = 72\) to solve the system:

• We first simplified the expression by dividing each side by 8. This reduced it to \(x^2 = 9\).
• The next step involved taking the square root of both sides to solve for \(x\), resulting in possible solutions of \(x = \pm 3 \).

When solving quadratic equations, remember that there can be two solutions due to the positive and negative square roots. Always consider both possibilities to fully capture the range of solutions a quadratic can provide in the context of a system of equations.

In the context of systems of equations, quadratic solutions are crucial, as they can uncover additional solutions that linear equations might miss.