Definition of a Mathematical Solution
In mathematics, a 'solution' refers to the value or set of values that makes a mathematical statement, such as an equation, inequality, or system of equations, true. When these values are substituted into the original statement, they satisfy the conditions specified, resulting in a correct mathematical assertion.
Solutions for Equations and Inequalities
For an equation (e.g., x + 5 = 10), the solution is the value(s) of the variable(s) that balance both sides of the equals sign. For an inequality (e.g., x > 3), the solution is typically a range of values that satisfy the condition, often represented as an interval or on a number line, indicating all possible values.
Example: Finding Solutions
Consider the equation 2x - 4 = 6. To find the solution, we isolate x by adding 4 to both sides (2x = 10) and then dividing by 2 (x = 5). Substituting 5 back into the equation (2*5 - 4 = 6) confirms it's true (10 - 4 = 6, which is 6 = 6). Thus, x=5 is the unique solution to this equation.
Importance in Problem Solving
Understanding solutions is fundamental to mathematical problem-solving across various fields, from basic algebra to advanced calculus and real-world applications. Finding solutions allows us to interpret and predict outcomes, model phenomena, and make informed decisions by identifying the specific conditions under which mathematical relationships hold true.