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A system of equations is a collection of two or more equations with the same set of variables. In this particular exercise, we have two equations:

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

Both equations have the variables \(x\) and \(y\). The goal is to find values for these variables that make both equations true simultaneously. Solving systems of equations is a fundamental part of algebra, and understanding how to manipulate these equations helps in tackling more complex mathematical problems. The addition method, sometimes called the elimination method, is particularly useful when the coefficients of one of the variables in both equations are equal and opposite. This allows for a straightforward elimination of one variable when the equations are added together.

Quadratic equations are equations of the form \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are constants. These are called quadratic because they contain terms up to the second degree, or squared terms. In this example, our equations contain quadratic terms:

• \(4x^2\) is present in both equations.
• \( - y^2 \) and \( + y^2 \) respectively in each equation.

Quadratic equations often have two solutions because squaring a number can produce the same result for both a positive and a negative root, like in the case of \(x^2 = 1\) leading to \(x = \pm 1\). Understanding and solving quadratic equations is a key skill because they appear frequently in various applications, from physics to finance.

The addition method for solving a system of equations involves adding two equations to eliminate one of the variables. In the given exercise, since both equations have \(4x^2\), adding them directly cancels out the \(y^2\) terms:

• Add \(4x^2 - y^2\) with \(4x^2 + y^2\) resulting in \(8x^2 = 8\).
• From here, solve for \(x^2\) by dividing both sides by 8 to get \(x^2 = 1\).
• Take the square root to find \(x = \pm1\).

Once \(x\) is found, substitute these values back into one of the original equations to solve for \(y\). This approach is an elegant and systematic way to find solutions, highlighting the power of algebraic manipulation in simplifying and solving systems of equations.