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 $6 .$ $\left\{\begin{aligned} \frac{3}{2} x-\frac{1}{3} y &=\frac{1}{2} \\ 2 x-\frac{1}{2} y &=-\frac{1}{2} \end{aligned}\right.$

$21-50=$ 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 $6 .$

$\left\{\begin{aligned} \frac{3}{2} x-\frac{1}{3} y &=\frac{1}{2} \\ 2 x-\frac{1}{2} y &=-\frac{1}{2} \end{aligned}\right.$

Key Concepts

Algebraic Manipulation

Effective algebraic manipulation, such as working with fractions, clearing denominators, and combining like terms, is crucial in solving systems of equations. These skills ensure that the equations are simplified correctly, minimizing errors and leading to the correct solution.

Types of Solutions in Linear Systems

Systems of linear equations can yield a unique solution, no solution, or infinitely many solutions. A unique solution occurs when the lines intersect at a single point, no solution when the lines are parallel, and infinitely many solutions when the lines are coincident.

System of Linear Equations

A system of linear equations consists of two or more equations with the same set of variables. The goal is to find the values of these variables that satisfy all equations concurrently, representing the point or points where the graphs of the equations intersect.

Elimination Method

The elimination method is a technique used to solve systems of linear equations by adding or subtracting the equations to cancel out one of the variables. This reduction simplifies the system to a single-variable equation, making it easier to solve.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This process reduces the system to a single equation with one variable and can simplify the solving process when one equation is easily manipulated.

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