key term - Dependent system

Definition

A dependent system is a system of equations in which the equations represent the same line, leading to an infinite number of solutions. This means that any point on the line is a solution, making the two equations essentially equivalent. When graphed, dependent systems overlap completely, which indicates that the two equations provide the same information about the relationship between the variables.

5 Must Know Facts For Your Next Test

1. Dependent systems arise when both equations in the system can be derived from each other through multiplication or addition of constants.
2. When solving a dependent system, you'll find that one equation can be expressed as a multiple or rearrangement of the other.
3. Graphing dependent systems will show two lines that lie exactly on top of each other, indicating their infinite points of intersection.
4. In applications, dependent systems often appear in real-world scenarios where multiple constraints yield the same relationship or outcome.
5. Identifying dependent systems can help simplify problem-solving, as recognizing the infinite solutions can eliminate unnecessary calculations.

Review Questions

• How can you determine if a given system of linear equations is dependent?
  • To determine if a system is dependent, you can manipulate both equations to see if one can be transformed into the other by applying operations such as addition, subtraction, or multiplication by a constant. If this holds true, then they represent the same line, confirming that there are infinitely many solutions. Additionally, graphing both equations will reveal if they overlap entirely, further indicating dependence.
• Discuss how dependent systems differ from independent and inconsistent systems in terms of solutions and graphing.
  • Dependent systems have an infinite number of solutions as their graphs overlap completely. In contrast, independent systems have exactly one solution where their graphs intersect at a single point. Inconsistent systems do not have any solutions since their graphs are parallel and do not intersect. Understanding these differences is crucial for analyzing and solving systems of equations effectively.
• Evaluate the implications of recognizing a dependent system in practical problem-solving scenarios.
  • Recognizing a dependent system in practical scenarios allows for a more efficient approach to problem-solving. Instead of treating each equation as separate constraints with distinct solutions, understanding their dependence means you can consolidate your analysis to focus on one relationship. This not only saves time but also helps clarify relationships within data sets or models that may initially seem complex due to multiple overlapping conditions.

Related terms

independent system: An independent system is a system of equations that has exactly one solution, meaning the graphs of the equations intersect at a single point.

inconsistent system: An inconsistent system is a system of equations that has no solutions, meaning the graphs of the equations are parallel and never intersect.

linear equations: Linear equations are equations of the first degree, which graph to straight lines and can be expressed in the form $$y = mx + b$$, where m is the slope and b is the y-intercept.