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.
9x - 5y = 1
-18x + 10y = 1

Verified step by step guidance

1. Start by writing down the system of equations clearly: \[9x - 5y = 1\] \[-18x + 10y = 1\]
2. Observe the coefficients of the variables in both equations. Notice that the second equation looks like it might be a multiple of the first. To check this, multiply the first equation by -2: \[-2 \times (9x - 5y) = -2 \times 1\] which gives \[-18x + 10y = -2\]
3. Compare the result from step 2 with the second equation in the system: The second equation is \[-18x + 10y = 1\], but after multiplying the first equation by -2, we got \[-18x + 10y = -2\]. Since the left sides are identical but the right sides are different, this means the two equations contradict each other.
4. Because the equations contradict, the system has no solution. This type of system is called an inconsistent system.
5. Therefore, conclude that the system is inconsistent and does not have any solutions.

Key Concepts

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, infinitely many points, or no solution.

Inconsistent and Dependent Systems

An inconsistent system has no solutions because the equations represent parallel lines that never intersect. A dependent system has infinitely many solutions because the equations represent the same line, meaning one equation is a multiple of the other.

Solving Systems by Substitution or Elimination

Methods like substitution or elimination help solve systems by isolating variables or combining equations to eliminate variables. Elimination involves adding or subtracting equations to remove one variable, simplifying the system to find solutions or determine consistency.