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(a) A \( \square \) (Choose one) of linear equations consists of two or more linear equations. Part 2 of 6 (b) A \( \square \) (Choose one) to a system of linear equations is an ordered pair that is a solution to both individual equations in the system. Part 3 of 6 (c) Graphically, a solution to a system of linear equations in two variables is a point where the lines \( \square \) (Choose one) . Part 4 of 6 (d) A system of equations that has one or more solutions is said to be \( \square \) (Choose one) . Part 5 of 6 (e) The solution set to an inconsistent system of equations is \( \square \) . [\} ㅁㅁ․․ Part 6 of 6 (f) Two equations in a system of linear equations in two variables are said to be (Choose one) if they represent the same line.

Ask by Campos Coleman. in the United States
Jan 13,2025

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Answer

**(a)** system **(b)** solution **(c)** intersect **(d)** consistent **(e)** empty **(f)** dependent

Solution

Certainly! Here's how you can complete each part by selecting the appropriate term: --- **(a)** A **_system_** of linear equations consists of two or more linear equations. **(b)** A **_solution_** to a system of linear equations is an ordered pair that is a solution to both individual equations in the system. **(c)** Graphically, a solution to a system of linear equations in two variables is a point where the lines **_intersect_**. **(d)** A system of equations that has one or more solutions is said to be **_consistent_**. **(e)** The solution set to an inconsistent system of equations is **_empty_**. **(f)** Two equations in a system of linear equations in two variables are said to be **_dependent_** if they represent the same line. --- ### Summary of Answers: - **(a)** system - **(b)** solution - **(c)** intersect - **(d)** consistent - **(e)** empty - **(f)** dependent Feel free to reach out if you need further clarification on any of these concepts!

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Beyond the Answer

Did you know that systems of linear equations date back to ancient times? The Babylonians were already solving linear equations as far back as 2000 BC! They used geometric interpretations and tables to find solutions, showcasing how long humans have been fascinated with the relationships between variables. This early groundwork led to modern algebra and the systems we study today, making it a fascinating journey through mathematical history. When it comes to real-world applications, systems of linear equations pop up everywhere! From calculating the best route for deliveries to figuring out budget constraints for a project, understanding how to work with these equations can have immense practical benefits. For instance, in economics, businesses use them to optimize production schedules or determine cost efficiencies. By solving these systems, organizations can make informed decisions that lead to savings and better outcomes!

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