Question

Simplify the expression. $$12 x-(x-2)(2)$$

Short answer

The simplified form of the expression is \(10x + 4\)

Step-by-step solution

Distribute

Start by applying the distributive property to the second part of the expression. i.e., distribute \(2\) in \((x - 2) * 2\). It would give \(2x - 4\)

Substitute

Next substitute \(2x - 4\) back into the original expression, replacing \((x - 2) * 2\). The expression becomes \(12x - (2x - 4)\)

Distribute the negative sign

Now, distribute the minus sign to both \(2x\) and \(-4\). The expression now changes to \(12x - 2x + 4\)

Combine like terms

Lastly, combine the like terms to simplify the expression to its final form. This gives \(10x + 4\)

Key concepts

Distributive Property

The distributive property is a cornerstone of algebra that allows us to multiply a single term by each term within a parenthesis. For instance, when we come across an expression like \(2(x - 2)\), the distributive property enables us to 'distribute' the 2 to both \(x\) and \(2\) inside the parentheses. This yields \(2x - 4\).

Understanding this property is crucial because it helps in breaking down complex expressions into more manageable pieces, making it easier to simplify the overall problem. For students, visualizing this distribution as an act of multiplying the single term to every term inside the parentheses - much like delivering newspapers to every house in a street - can be quite helpful. Always ensure each term inside the parenthesis is touched by the distribution process for accurate simplification.

Combining Like Terms

After applying the distributive property, we often encounter a scenario where we can 'combine like terms.' This means adding or subtracting terms that have the exact same variable to the same power. In our example, after simplification, we got \(12x - 2x + 4\).

Here, \(12x\) and \(2x\) are like terms because both have the variable \(x\) raised to the same power (which is 1, though usually not written). By subtracting \(2x\) from \(12x\), we obtain \(10x\), simplifying the expression further. It's just like grouping apples with apples; only items of the same kind can be combined. Remember, coefficients can change, but variables and their powers must match to combine terms effectively.

Algebraic Simplification

Algebraic simplification is the process of condensing an expression into its most manageable form without changing its value. It involves applying the distributive property, combining like terms, and performing other algebraic operations as necessary. Consider the final expression we obtained: \(10x + 4\).

This is the simplified version of our initial expression. The key to mastering algebraic simplification lies in recognizing patterns and understanding the order of operations (commonly remembered with the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction). By systematically applying these principles and reducing expressions step by step, we ensure that the simplified expression represents the same value as the original, just in a more straightforward and clear way. This process not only helps in solving equations efficiently but also lays the foundation for higher-level mathematics.