Ecuaciones de Primer Grado

Study Notes

Equations of the First Degree

Equations of the first degree are algebraic equations where the highest exponent of the variable is 1. They are also known as linear equations.
The general form of a linear equation in one variable (x) is ax + b = 0, where 'a' and 'b' are constants, and 'a' ≠ 0.
To solve a linear equation, the goal is to isolate the variable (x) on one side of the equation.

Solving Linear Equations

Combining Like Terms: Simplify both sides of the equation by combining any terms that have the same variable and exponent.
Adding or Subtracting: Add or subtract the same value from both sides of the equation to isolate the variable term.
Multiplication or Division: If the variable is multiplied or divided by a constant, perform the opposite operation on both sides of the equation to isolate the variable.
Distributive Property: If parentheses are present, apply the distributive property to eliminate them before proceeding with the other steps.
Example: Solve 2x + 5 = 11; Subtract 5 from both sides: 2x = 6; Divide both sides by 2: x = 3

Types of Solutions

Unique Solution: A linear equation has one unique solution if the coefficient of the variable (x) is not zero.
No Solution: A linear equation has no solution if the variable terms cancel out and the constant terms are unequal. For example, 2x + 3 = 2x + 7.
Infinite Solutions: A linear equation has infinite solutions if both sides of the equation are identical when simplified. For example, 2(x + 1) = 2x + 3 (simplifies to 2x + 2 = 2x + 2).
Important Note: Equations can involve fraction coefficients. Multiplying both sides by the least common denominator (LCD) of the fractions can make solving easier, removing the fractions.

Applications of Linear Equations

Linear equations are used in numerous applications, including:
- Geometry: Calculating areas, perimeters, and angle measures.
- Business: Modeling costs, revenues, and profits.
- Physics: Describing motion, force, and work.
- Everyday Problems: Solving practical situations involving unknown values, such as calculating ingredient amounts for a recipe or time needed to complete a task.

Key Concepts

Variable: A symbol (usually a letter) that represents an unknown numerical value.
Constant: A fixed numerical value.
Equation: A mathematical statement that shows the equality of two expressions.
Solution: The value of the variable that makes the equation true.
Equivalent Equations: Equations that have the same solution.

Description

Este cuestionario se centra en las ecuaciones de primer grado, también conocidas como ecuaciones lineales. Explora la forma general de estas ecuaciones, así como los métodos para resolverlas paso a paso. Utiliza ejemplos prácticos para aplicar el concepto.