Question

Simplify the expression. Assume all variables are positive. \(\frac{x+3 x}{2}\)

Short answer

The simplification of the given expression \(\frac{x+3 x}{2}\) is \(2x\).

Step-by-step solution

Identify the terms in the expression

The expression given is \(\frac{x+3 x}{2}\). In this expression, our main focus will be the terms that appear in the numerator, which are \(x\) and \(3x\).

Apply the distributive property

We can add the terms in the numerator because they are like terms - they both contain the same variable, \(x\), to the same power of 1. Adding \(x\) and \(3x\), we get \(4x\). So, the expression becomes \(\frac{4x}{2}\).

Divide the numerator by the denominator

Now, divide \(4x\) by 2. The result of the division is \(2x\).

Key concepts

Distributive Property

The distributive property is a cornerstone in algebra that allows us to multiply a single term by each term within a parenthesis in an expression. Essentially, it lets you 'distribute' a number or variable across the terms inside the brackets. Formally, it states that for any numbers a, b, and c, the equation \( a(b + c) = ab + ac \) holds true.

When simplifying algebraic expressions, if we encounter a situation where a single term is being multiplied by an expression inside a parenthesis, we can apply the distributive property to simplify the expression. This is particularly useful when combining like terms or when an expression looks complex and distributing helps in making it more manageable.

For example, if we have the expression \( 3(2 + x) \), we can use the distributive property to multiply 3 by each term inside the parenthesis, resulting in \( 6 + 3x \).

Like Terms

In algebra, 'like terms' refers to terms that have the same variables raised to the same power. Even if their numerical coefficients are different, terms are 'like' if the variable parts match. This is why we can only simplify algebraic expressions by combining like terms.

The process of simplification involves adding or subtracting coefficients while keeping the variable part the same. Think of it like grouping similar items; you wouldn't mix apples with oranges when counting, right? The same principle applies when combining like terms in algebra.

For instance, \( x \) and \( 3x \) are like terms because they both contain the variable \( x \) to the same power, which is 1 by default when no other exponent is shown. We combine their coefficients to get \( 4x \) when simplifying expressions.

Numerical Coefficient

The numerical coefficient of a term in an algebraic expression is the number that multiplies the variable or variables. It's an important concept that gives us insight into how the value of the term increases or decreases in response to the variable it accompanies.

In the term \( 3x \), \( 3 \) is the numerical coefficient, indicating that \( x \) is multiplied by \( 3 \). Coefficients can be positive, negative, or even fractions, giving us a wide range of values that a term can take based on the variable's value.

The numerical coefficient is critical when we're simplifying expressions, especially when like terms are involved. Without this understanding, it would be difficult to correctly combine or separate terms during the simplification process.