Solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.
${\begin{matrix} x + 4 y = 14 \\ 2 x - y = 1 \end{matrix}$

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Identify the system of equations you need to solve. Typically, this will be two or more equations involving the same variables.

Choose a method to solve the system: substitution, elimination, or graphing. For algebraic clarity, substitution or elimination is often preferred.

If using substitution, solve one equation for one variable in terms of the other(s), then substitute this expression into the other equation(s) to reduce the system to one equation with one variable.

If using elimination, manipulate the equations (by multiplying if necessary) so that adding or subtracting them eliminates one variable, allowing you to solve for the remaining variable.

After finding the value(s) of the variable(s), substitute back into one of the original equations to find the other variable(s). Then, analyze the system: if you get a true statement with no variables, the system has infinitely many solutions; if you get a false statement, the system has no solution; otherwise, write the solution set in set notation, for example, $\{(x,y) \mid x = a, y = b\}$.

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Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. Solutions can be a single point, no solution, or infinitely many solutions depending on the relationships between the equations.

Methods for Solving Systems

Common methods to solve systems include substitution, elimination, and graphing. Substitution involves solving one equation for a variable and substituting into another. Elimination adds or subtracts equations to eliminate a variable. Graphing visualizes solutions as intersection points of lines.

Types of Solutions and Set Notation

Systems can have one solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (coincident lines). No solution means the system is inconsistent. Solutions are expressed using set notation, such as {(x, y) | x = 2, y = 3} for a single solution or {(x, y) | y = 2x + 1} for infinite solutions.

Solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.{x+4y=142x−y=1\(\begin{cases}\) x + 4y = 14 \\ 2x - y = 1 \(\end{cases}\)