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When simplifying algebraic expressions, it's important to know how numbers can be manipulated without changing their sum, which is where the associative property of addition comes into play. This property states that the way in which numbers are grouped in an addition operation does not affect the sum. For example, if you have the expression \(a + (b + c)\), you can regroup the numbers to look like \( (a + b) + c\) without changing the result.

Applying this to the exercise \( (12 x - 3)+(3 x - 4) \), we can remove the parentheses to combine terms more easily. The associative property assures us that rearranging these groups does not alter the amount they sum up to. Understanding this fundamental property enables students to tackle more complex algebra without hesitation.

Simplifying algebraic expressions is a cornerstone of working in algebra. To simplify an expression means to make it as basic or streamlined as possible, usually by combining like terms and using algebraic properties effectively. Like terms are terms that have the same variable raised to the same power, such as \(12x\) and \(3x\).

In our example problem, we use the initial step to remove the parentheses, resulting in \(12x - 3 + 3x - 4\). Then, we combine the like terms \(12x\) and \(3x\) to form \(15x\), and do the same with the constant terms \( -3\) and \( -4\) to get \( -7\). Simplification is important for understanding and solving algebraic equations and expressions, as it reduces complexity and makes it easier to see relationships between variables and numbers.

Elementary algebra is the branch of mathematics that introduces students to the concepts of variables and constants and the use of algebraic expressions and equations. This type of algebra serves as the foundation for more advanced mathematical studies. Solving an algebraic expression, such as \(12x - 3 + 3x - 4\), involves understanding how to manipulate variables and constants to simplify or solve for unknown values.

It's essential to grasp the basics of elementary algebra to move on to more intricate problems. In our given expression, the ability to identify and combine like terms, and apply the associative property, shows a practical utilization of elementary algebra. Mastery of these skills is crucial for academic advancement in mathematics.