21-50= Solving a System of Equations Solve the system, or...

$21-50=$ Solving a System of Equations Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example $6 .$

$x+y=7$
$2 x-3 y=-1$

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Key Concepts

Systems of Linear Equations

This concept involves a collection of two or more linear equations that share the same set of variables. The solution to the system is any set of values for these variables that makes every equation in the system true at the same time. Systems of linear equations are fundamental in algebra and appear in various contexts in science and engineering.

Methods for Solving Systems

There are several techniques to solve systems of linear equations, including the substitution method and the elimination method. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation(s), while the elimination method involves combining the equations in a way that eliminates one variable, allowing for the other variable(s) to be solved directly.

Consistency and Uniqueness of Solutions

Systems of equations can be classified based on their solutions. A system is consistent if it has at least one solution and inconsistent if it has none. Consistent systems can further be divided into those with a unique solution or infinitely many solutions, depending on whether the equations refer to the same line or intersect at a single point.

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