true statement
[true statement|true statement] The true statement 0 = 0 indicates a dependent system with infinitely many solutions.
Definition
true statement The true statement 0 = 0 indicates a dependent system with infinitely many solutions. This equation lacks real values of x that satisfy it, as any power of a positive number remains positive. The statement reflects a system where multiple solutions exist without contradiction. It highlights the dependency between equations in the system. The concept underscores the presence of infinitely many valid solutions within the system.
Causes
true statement Substitute the solution into the original equation to make sure the result is a true statement. There is no real value of x that will make the equation a true statement because any power of a positive number is positive. This confirms that the equation cannot yield a true statement for any real x.
Effects
true statement Substituting the solution into the original equation ensures the result is a true statement. The equation's validity depends on the power of a positive number remaining positive. No real value of x can make the equation a true statement due to this property.
Examples
true statement The equation 2 x + 1 = 7 has the solution 3. Substituting 3 for x yields the true statement 2 ( 3 ) + 1 = 7. This demonstrates how substituting the solution into the equation produces a valid result.
Dependent System
true statement The true statement 0 = 0 indicates a dependent system with infinitely many solutions. This system is characterized by having many solutions rather than a unique solution. A dependent system is defined by its equations being multiples of each other, leading to this outcome.
Effects on Original Equation
Substituting the solution into the original equation confirms that true statement holds true, thereby validating the solution's accuracy.
Real Value
true statement The real value of x cannot satisfy the equation because any power of a positive number remains positive, making the equation invalid.