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 $6 .$ $\left\{\begin{aligned} \frac{3}{4} x+\frac{1}{2} y &=5 \\-\frac{1}{4} x-\frac{3}{2} y &=1 \end{aligned}\right.$

$21-50=$ 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 $6 .$

$\left\{\begin{aligned} \frac{3}{4} x+\frac{1}{2} y &=5 \\-\frac{1}{4} x-\frac{3}{2} y &=1 \end{aligned}\right.$

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Key Concepts

System of Linear Equations

A system of linear equations consists of multiple linear equations that share the same set of variables. The goal is to find the values of these variables that satisfy all the equations simultaneously. These systems form the basis for many techniques in algebra and can be solved by various methods, depending on the number of equations and variables involved.

Elimination Method

The elimination method is a technique used to solve systems of linear equations by adding or subtracting equations to cancel one of the variables. By strategically eliminating variables, the system is reduced to a simpler equation with one variable, making it straightforward to solve and then back-substitute to find the other variable(s).

Consistency and Types of Solutions

Systems of linear equations can be classified as consistent or inconsistent. A consistent system has at least one solution, which may be unique or infinitely many (in the case of dependent equations). An inconsistent system has no solution because the equations contradict each other, meaning that no single set of variable values satisfies all the equations at the same time.

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