Number of Solutions of the System | Active Summary

Objectives

1. Identify and differentiate systems of equations with a unique solution, infinite solutions, and no solution.

2. Develop analytical skills to solve systems of equations using methods such as substitution, linear combination, and graphs.

3. Apply the acquired knowledge to model and solve everyday problems, such as situations of speed, production, and cost.

Contextualization

Did you know that systems of equations are used in many fields beyond mathematics? For example, in engineering, they are fundamental for designing bridges and buildings, ensuring they are safe and stable. Likewise, in economics, they help optimize production and understand the market. Mastering these concepts is not only an academic necessity but an essential tool for solving real-world problems!

Important Topics

Unique Solution

A system of equations has a unique solution when the lines that represent the equations intersect at a single point. This means there is a unique set of values for the variables that satisfies all the equations simultaneously, forming an intersection point. This case is common in problems involving the intersection of two independent variables.

• Graphical representation: In the Cartesian plane, the unique solution is the intersection point of the two lines that represent the equations. This point is the solution to the system.

• Substitution method: Can be used to solve systems with a unique solution by substituting one variable in an equation with the value of another found in the other equation.

• Practical application: Used in situations such as calculating expenses and revenues where the two quantities depend on a common variable.

Infinite Solutions

A system of equations has infinite solutions when the equations are represented by coincident lines, meaning the equations are multiples of each other. This indicates that all solutions of the first equation are also solutions of the second, and vice versa. This case occurs when the equations are essentially the same line.

• Coincident graphs: In the Cartesian plane, coincident lines completely overlap, meaning all points on the line are solutions.

• Solution verification: All points on the line are valid solutions, which is verified by substituting the values into the equations.

• Utility: Can occur in situations where two equations represent the same information differently, such as measurements converted from one unit to another.

No Solution

A system of equations has no solution when the equations are parallel and do not intersect at any point. This indicates that there are no common values that satisfy all equations simultaneously. This case is common when the two equations represent lines with the same slope but intercept different values on the y-axis.

• Parallel graphs: In the Cartesian plane, parallel lines do not intersect, indicating the absence of a solution.

• Coefficient analysis: If the slopes are equal and the constant terms are different, there is no solution.

• Practical relevance: Can occur in situations of constraint where the equations represent limits that cannot be satisfied simultaneously.

Key Terms

• System of Equations: A set of equations that share the same variables and that, together, can be solved to find the values of those variables.

• Solution of a System: The set of values for the variables that satisfy all the equations of the system simultaneously.

• Intersection: Common point where two or more lines intersect in the Cartesian plane, indicating the solution of a system of equations.

To Reflect

• How can understanding the number of solutions in systems of equations help in solving everyday problems, such as personal financial planning?

• Why is it important to verify the solutions found in systems of equations, especially in real contexts, before applying them?

• In what way can the ability to model situations with systems of equations impact choices and decisions in professions such as engineering and economics?

Important Conclusions

• During our mathematical journey, we explored the different types of solutions in systems of equations: unique solution, infinite solutions, and no solution. We understood how to apply graphical and algebraic methods to solve these systems, which is essential for various practical applications.

• The ability to model real situations using systems of equations is crucial, as it allows for informed decision-making and solving complex problems in areas such as engineering, economics, and computer science.

• Understanding and applying the concept of systems of equations is not only an academic skill but a valuable tool in everyday life and various careers, helping to optimize processes and make data-driven decisions.

To Exercise Knowledge

1. Create a system of equations that represents the situation of dividing a budget among different fixed and variable expenses. Solve it and interpret the solution. 2. Use a system of equations to model the situation where two cars travel at different speeds and find out when they will meet if they depart from different points. 3. Develop a system of equations that represents the mixture of two products with different costs and determine the proportion that minimizes the total cost.

Challenge

System Detective Challenge: Imagine you are a detective and you received a case that requires using systems of equations to decipher a code. Create a system of equations that represents different coded letters of the alphabet and decipher a secret message. Present your solution in a small report explaining the process and logic used.

Study Tips

• Practice drawing graphs of equations to better visualize the solutions. Use online tools like Geogebra to experiment and understand the relationships between the equations.

• Try solving systems of equations in different ways (substitution, linear combination, graphs) to reinforce understanding and flexibility in problem-solving.

• Challenge yourself to create your own systems of equations problems based on situations of your interest, such as sports, music, or games, to see mathematics in action in your daily life.