Solve each system of equations. State whether it is an inconsistent system or has infinitely many solutions. If a system has infinitely many solutions, write the solution set with x arbitrary. See Examples 3 and 4. 9x - 5y = 1 -18x + 10y = 1

Key Concepts

Systems of Equations

A system of equations consists of two or more equations that share the same variables. The goal is to find values for these variables that satisfy all equations simultaneously. Solutions can be unique, none, or infinitely many, depending on the relationships between the equations.

Inconsistent Systems

An inconsistent system of equations is one that has no solution. This occurs when the equations represent parallel lines that never intersect. In such cases, the equations contradict each other, indicating that there is no set of values for the variables that can satisfy all equations in the system.

Infinitely Many Solutions

A system has infinitely many solutions when the equations represent the same line, meaning they overlap completely. This typically occurs when one equation is a scalar multiple of the other. In this case, the solution set can be expressed with one variable as arbitrary, indicating that there are countless combinations of values that satisfy the equations.