Solve the following system for x and y in terms of a and b...

Solve the following system for $x$ and $y$ in terms of $a$ and $b$ where $a$ and $b$ are nonzero and $a \neq b$
$$
\left\{\begin{array}{l}
\frac{a}{b x}+\frac{b}{a y}=a+b \\
\frac{b}{x}+\frac{a}{y}=a^{2}+b^{2}
\end{array}\right.
$$

Key Concepts

System of Equations

A system of equations consists of multiple equations that are solved together to find common solutions for the variables. This concept involves strategies to ensure every equation's requirements are met simultaneously and is foundational in algebra.

Rational Equations

Rational equations involve variables that appear in the denominators of fractions. Solving them often requires techniques to clear the fractions, such as multiplying through by a common denominator, while carefully considering restrictions that may arise from values that make a denominator zero.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations using operations like cross-multiplication, factorization, and combining like terms. This skill is critical to transform complicated expressions into simpler forms that are more manageable for solving.

Elimination and Substitution Methods

Both elimination and substitution are key techniques for solving systems of equations. The elimination method involves adding or subtracting equations to remove one variable, while the substitution method involves solving one equation for one variable and then replacing that variable in the other equation.

Parameter Dependency and Domain Restrictions

When equations include parameters along with variables, it is important to consider how the solutions depend on these parameters and to observe any conditions imposed on them, such as nonzero values or distinct values. This ensures that the derived solutions are valid under the specified constraints.

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