Solve each system for x and y, expressing either value in...

Solve each system for $x$ and $y,$ expressing either value in terms of a or $b$, if necessary. Assume that $a \neq 0$ and $b \neq 0.$
$\left\{\begin{array}{r}{5 a x+4 y=17} \\ {a x+7 y=22}\end{array}\right.$

Key Concepts

Systems of Linear Equations

A system of linear equations involves two or more linear equations with the same set of unknowns. The goal is to find all values of the variables that satisfy every equation at the same time. These systems can consist entirely of constant coefficients or include parameters, and they form the basis for many methods in algebra.

Elimination Method

The elimination method is a technique for solving systems of linear equations by combining the equations to cancel out one of the variables. This is achieved by multiplying the equations by appropriate factors and then adding or subtracting them, reducing the system to a single equation in one variable, which can then be solved directly.

Substitution Method

The substitution method addresses a system by first solving one of the equations for one variable in terms of the others. This expression is then substituted into the remaining equations. This method reduces the number of variables and simplifies the system, making it easier to solve sequentially.

Handling Parameters and Nonzero Constraints

When systems of linear equations include parameters, such as coefficients that are not specified by numerical values, it is important to consider any restrictions on these parameters. Nonzero constraints, for example, ensure that division by zero does not occur during the solving process and that unique solutions exist. Understanding these constraints is critical for expressing the solution accurately in terms of the parameters.

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