In Exercises 25–35, solve each system by the method of your choice. This is a piecewise function, refer to textbook problem.

Verified step by step guidance

1. Identify the given piecewise function and the system of equations associated with it. A piecewise function is defined by different expressions depending on the value of the input variable. Carefully note the conditions for each piece of the function.
2. Determine which method you want to use to solve the system of equations. Common methods include substitution, elimination, or graphing. Choose the method that seems most efficient based on the structure of the equations.
3. For each piece of the piecewise function, solve the corresponding equation or inequality. Ensure that you respect the domain restrictions provided for each piece of the function.
4. If using substitution or elimination, simplify the equations step by step. For substitution, solve one equation for one variable and substitute it into the other equation. For elimination, add or subtract equations to eliminate one variable.
5. After solving for the variables, verify that the solution satisfies the conditions of the piecewise function. Check that the solution lies within the correct domain for the piece of the function being used.

Key Concepts

Systems of Equations

A system of equations consists of two or more equations with the same set of variables. The goal is to find the values of the variables that satisfy all equations simultaneously. Common methods for solving systems include substitution, elimination, and graphing. Understanding how to manipulate and solve these equations is crucial for finding solutions.

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. Each 'piece' of the function applies to a specific interval of the domain. To solve problems involving piecewise functions, one must identify which expression to use for a given input and ensure continuity and correctness across the defined intervals.

Graphical Interpretation

Graphical interpretation involves visualizing equations and functions on a coordinate plane. For systems of equations, the solution can often be found at the intersection points of the graphs. Understanding how to sketch and analyze these graphs helps in identifying solutions and understanding the behavior of piecewise functions across different intervals.