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Mathematics: Equations

Mathematics: Equations

Definition

Definition
An equation is a mathematical equality that contains at least one unknown. The solution of the equation is the value or values of the unknown that make the equality true.
Equations are a fundamental concept in mathematics, used in many fields such as algebra, geometry, analysis, and probability. They are used to solve problems by finding the values of unknowns that satisfy a certain condition. Equations are often represented in the form of an equality between two expressions, with an equal sign in the middle. For example, the following equation:
2x + 3 = 7
contains the unknown x and asks to find what value of x makes the equality true.

Definition

Solutions
The solutions of an equation can take different forms depending on the type of equation. Some equations may have a unique solution, others may have multiple solutions or even no solution.
For example, the equation 2x + 3 = 7 has a unique solution, which is x = 2. By substituting this value into the equation, we obtain:
2(2) + 3 = 7
4 + 3 = 7
7 = 7
The equality holds, so the value x = 2 is indeed a solution of the equation.
Some equations may also have complex solutions, which are imaginary numbers. For instance, the quadratic equation x² + 1 = 0 has no real solution, but it does have a complex solution x = i, where i is the imaginary unit.

Definition

Methods of Resolution
There are different methods for solving equations depending on their type. Some equations can be solved by applying mathematical operations to both sides of the equality, while others require the use of specific formulas or numerical methods.
Among the common resolution methods, we find:
  • The substitution method: by replacing one expression with another at each step, the equation is simplified until only one unknown remains.
  • The elimination method: by adding or subtracting equations to eliminate one unknown at each step, the equation is reduced to a form where only one unknown remains.
  • The Cartesian product method: by using value tables to test different combinations until finding those that satisfy the equation.
  • The graphical method: by graphically representing the equation on a Cartesian plane, one finds the intersection points between the curve and the x-axis that represent the solutions of the equation.
    • The resolution method depends on the type of equation and the mathematical tools available to us. It is important to choose the appropriate method to obtain the correct solutions.

    To Remember:

    Summary:

    Equations are mathematical equalities that contain one or more unknowns. The solutions of an equation are the values of the unknown that make the equality true. There are different methods for solving equations depending on their type. Some equations have a unique solution, others have multiple solutions, or none at all. Equations are used in many fields of mathematics to solve problems and model real situations.


    Mathematics: Equations

    Mathematics: Equations

    Definition

    Definition
    An equation is a mathematical equality that contains at least one unknown. The solution of the equation is the value or values of the unknown that make the equality true.
    Equations are a fundamental concept in mathematics, used in many fields such as algebra, geometry, analysis, and probability. They are used to solve problems by finding the values of unknowns that satisfy a certain condition. Equations are often represented in the form of an equality between two expressions, with an equal sign in the middle. For example, the following equation:
    2x + 3 = 7
    contains the unknown x and asks to find what value of x makes the equality true.

    Definition

    Solutions
    The solutions of an equation can take different forms depending on the type of equation. Some equations may have a unique solution, others may have multiple solutions or even no solution.
    For example, the equation 2x + 3 = 7 has a unique solution, which is x = 2. By substituting this value into the equation, we obtain:
    2(2) + 3 = 7
    4 + 3 = 7
    7 = 7
    The equality holds, so the value x = 2 is indeed a solution of the equation.
    Some equations may also have complex solutions, which are imaginary numbers. For instance, the quadratic equation x² + 1 = 0 has no real solution, but it does have a complex solution x = i, where i is the imaginary unit.

    Definition

    Methods of Resolution
    There are different methods for solving equations depending on their type. Some equations can be solved by applying mathematical operations to both sides of the equality, while others require the use of specific formulas or numerical methods.
    Among the common resolution methods, we find:
  • The substitution method: by replacing one expression with another at each step, the equation is simplified until only one unknown remains.
  • The elimination method: by adding or subtracting equations to eliminate one unknown at each step, the equation is reduced to a form where only one unknown remains.
  • The Cartesian product method: by using value tables to test different combinations until finding those that satisfy the equation.
  • The graphical method: by graphically representing the equation on a Cartesian plane, one finds the intersection points between the curve and the x-axis that represent the solutions of the equation.
    • The resolution method depends on the type of equation and the mathematical tools available to us. It is important to choose the appropriate method to obtain the correct solutions.

    To Remember:

    Summary:

    Equations are mathematical equalities that contain one or more unknowns. The solutions of an equation are the values of the unknown that make the equality true. There are different methods for solving equations depending on their type. Some equations have a unique solution, others have multiple solutions, or none at all. Equations are used in many fields of mathematics to solve problems and model real situations.