Solve each system by the method of your choice.
$\{\begin{array}{c}5y=x^2-1\\ x-y=1\end{array}$

Verified step by step guidance

1. Identify the system of equations or inequalities given in the problem. Since it mentions a piecewise function, carefully write down each piece of the function along with its domain restrictions.
2. For each piece of the piecewise function, set up the corresponding equation or inequality that applies within its domain. This will help you understand which part of the system to solve in each interval.
3. Choose a method to solve the system, such as substitution, elimination, or graphing. For piecewise functions, it is often helpful to solve each piece separately within its domain and then combine the solutions.
4. Solve each equation or inequality step-by-step within the specified domain. Make sure to check that your solutions satisfy the domain restrictions of each piece of the function.
5. After finding solutions for each piece, combine them to write the complete solution to the system. Verify your answers by substituting them back into the original piecewise function to ensure they satisfy all parts of the system.

Key Concepts

Systems of Equations

A system of equations consists of two or more equations with the same set of variables. Solving the system means finding values for the variables that satisfy all equations simultaneously. Methods include substitution, elimination, and graphing.

Piecewise Functions

A piecewise function is defined by different expressions depending on the input value's domain. Understanding how to evaluate and interpret these functions is essential, especially when solving systems involving piecewise definitions.

Solving Systems Involving Piecewise Functions

When solving systems with piecewise functions, you must consider each piece separately within its domain. This often requires checking solutions against domain restrictions to ensure they are valid for the corresponding piece.