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 $6 .$

$\frac{1}{2} x+\frac{1}{3} y=2$
$\frac{1}{5} x-\frac{2}{3} y=8$

Key Concepts

Ordered Pair Representation

When the solution to a system of two variables is found, it is commonly expressed in the form of an ordered pair. This notation clearly indicates the values of the variables and is especially useful when representing either a single unique solution or the general form of infinitely many solutions.

Consistency and Uniqueness of Solutions

In a system of linear equations, the analysis often focuses on whether the system is consistent (has at least one solution) or inconsistent (has no solution). Additionally, when a system is consistent, it may have a unique solution or infinitely many solutions. Understanding these conditions is crucial for determining the nature of the solutions and selecting an appropriate method for solving the system.

Systems of Linear Equations

A system of linear equations consists of two or more equations involving the same set of variables. The purpose is to find values for these variables that satisfy all the equations simultaneously. This concept is fundamental in algebra and appears in various applications in science and engineering.

Methods for Solving Systems

There are several techniques for solving systems of linear equations, including the substitution method and the elimination method. The substitution method involves solving one of the equations for one variable and substituting that expression into the other equation, while the elimination method involves adding or subtracting equations to cancel out one of the variables. Both methods simplify the system into a form where the solution becomes clear.

Clearing Fractions

Clearing fractions is a strategy used to simplify equations that contain fractional coefficients. By multiplying both sides of the equation by an appropriate common denominator, the fractions are removed, making it easier to perform further algebraic manipulations and solve the equation.

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