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A quadratic equation is a polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( x \) is an unknown variable. In the context of solving systems of equations using the substitution method, quadratic equations often arise when substituting one variable in terms of another leads to a quadratic expression.

The equation \( 2x^2 - x - 6 = 0 \) from the problem is a perfect example. When solving quadratic equations, there are a few methods you can use:

• Factoring, if the quadratic can be factored.
• Completing the square.
• Using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

The quadratic formula is especially useful for equations that are not easily factorable. Always remember to check your solutions by substituting them back into the original equations, ensuring they satisfy each condition.

A system of equations is essentially a set of two or more equations that share the same set of variables. The objective is to find values for the variables that satisfy all equations in the system simultaneously. In such systems:

Each equation provides unique information about the relationships between the variables. When one equation is manipulated, the relationships can be reformed into a simplified version to make solving easier.

In the given example, the system of equations is:

• \( xy = 6 \)
• \( 2x - y = 1 \)

The goal is to find the values of \( x \) and \( y \) that make both equations true. By using the substitution method, one variable is isolated in one equation and replaced in another, translating the system of equations into a single equation.

Isolating variables is a crucial step in solving a system of equations, particularly when using the substitution method. It involves expressing one variable in terms of the others, allowing it to be replaced in another equation.

In solving the provided system, we start by isolating \( y \) in the first equation: \( xy = 6 \). Upon rearranging, we get \( y = \frac{6}{x} \). This isolated expression can then be substituted back into the second equation \( 2x - y = 1 \), resulting in an equation in terms of \( x \) alone. By solving this, we reduce the number of unknowns, facilitating the solution process further.

Isolating variables efficiently reduces computational complexity and makes it straightforward to solve more complex systems in algebra. Properly isolating variables is the key to using substitution effectively, ensuring each variable is expressed clearly for further operations.