Problem 51 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 3(x+5)+2 x $$

Step-by-Step Solution

Answer

Answer: The simplified form of the given expression is $5x + 15$.

Step 1: Distribute the constant

Distribute the constant 3 across the terms inside the parentheses by multiplying each term within the parentheses by 3: $$ 3(x+5)+2 x = 3x +15 +2x $$

Step 2: Combine like terms

Now combine the like terms, in this case, the terms involving x, which are 3x and 2x: $$ 3x +15 +2x = (3x+2x) +15 $$ $$ = 5x +15 $$ The simplified expression is: $$ 5x + 15 $$

Key Concepts

Distributive Property

When simplifying algebraic expressions, the distributive property is your key to unlocking grouped terms. Imagine being given a basket containing different fruits, and your task is to hand out the fruits equally to your friends. In a similar way, the distributive property allows you to 'distribute' a number or a variable across terms inside parentheses.

Here's how it works: When you have an expression such as \(a(b+c)\), you apply the distributive property by multiplying the outer term \(a\) with each of the terms inside the parentheses: \(ab + ac\). In our exercise example, \(3(x + 5) + 2x\), you multiply 3 by both \(x\) and 5, which results in \(3x + 15\). The distributive property ensures that each term within the parentheses is accounted for in the multiplication process, making it a crucial first step in the simplification of algebraic expressions.

Combining Like Terms

After utilizing the distributive property, you may find your algebraic expression filled with terms that seem similar. This is where the concept of 'combining like terms' comes into play. Just as you would sort and group the same colored socks together when organizing your drawer, combining like terms means lumping together the terms in an algebraic expression that have the same variable raised to the same power.

In the expression \(3x + 15 + 2x\), the terms \(3x\) and \(2x\) are considered like terms because both have the variable \(x\) raised to the same power, which is 1 (though the power of 1 is not always written). You combine these terms by adding their coefficients, the numerical part in front of the variable: \(3x + 2x = 5x\). Combining like terms streamlines your expression, making it neater and easier to handle in subsequent calculations.

Algebraic Simplification

With the groundwork laid by the distributive property and combining like terms, you are now ready for the final step of algebraic simplification. This is where you polish your expression, ensuring it's as neat and concise as possible. Algebraic simplification can include a variety of actions such as combining like terms, factoring, canceling, and applying algebraic rules to achieve the simplest form of an expression.

In the context of our given problem, once you've distributed and combined like terms, you are left with \(5x + 15\). This final form is the essence of algebraic simplification: it's the version of your expression that conveys the same information as the original, but in the most straightforward manner. Simplified expressions are highly desirable in algebra because they make subsequent mathematical operations and understanding the behavior of functions far easier.