ECUACIONES DE 1° GRADO (CASOS)

Introduction to Real Numbers and First-Degree Equations

What is an Equation?

• An equation is defined as an equality, which can be illustrated through examples. The speaker uses the scenario of a man walking at a constant speed towards his destination to explain the components involved: speed, time, and distance.

Mathematical Representation

• To analyze real-world situations mathematically, we use mathematical language. The relationship between space, speed, and time is expressed as textspace = textspeed times texttime , which relates to uniform linear motion.

Understanding First-Degree Equations

• The focus shifts to first-degree equations in the form ax = b , where a and b are real numbers. Different scenarios arise based on the values of a and b .

Analyzing Cases for First-Degree Equations

Case 1: Unique Solution

• When a neq 0, the equation has one unique solution known as the root or solution of the equation. This root satisfies the equation and can be calculated using x = b/a .

Example Calculation

• An example is provided with an equation involving fractions:

• The speaker demonstrates how to manipulate terms to isolate variables leading to solutions.

Case 2: Infinite Solutions

• If both a = 0 and b = 0, any value for x will satisfy the equation (e.g., 0x = 0). This results in infinite solutions represented by all real numbers.

Example of Infinite Solutions

• A second example illustrates this case where simplification leads back to zero on both sides of an equation confirming that every real number is a solution.

Conclusion on First-Degree Equations

Case Number 3: Understanding Linear Equations

Conditions for Case Number 3

• The case arises when the value of a is not equal to zero and b equals zero, leading to the linear equation format 0x = b .

• According to the property of absorption, any number multiplied by zero results in zero; thus, this equation cannot hold true if b neq 0 .

Implications of the Equation

• Since no number can satisfy the condition where a non-zero number is multiplied by zero, this proposition is deemed false.

• Consequently, it is concluded that the equation has no solutions, resulting in an empty solution set.

Finding Omega: Step-by-Step Calculation

Setting Up the Equation

• The expression involves calculating values for Omega , starting with manipulating terms such as -2(1 - Omega) = 5Omega/2 - 3 .

• Careful attention is given to parentheses as they affect subsequent operations within the expression.

Simplifying Expressions

• By multiplying through by denominators and simplifying fractions, we arrive at expressions like -4(1 - Omega)/2 = (5Omega - 6)/2 .

• Further simplification leads to an expression indicating that 0Omega = -2 , which ultimately reveals a contradiction.

Conclusion on Omega's Value