Question

Explain to a friend how the Distributive Property is used to justify the fact that \(2 x+3 x=5 x\).

Short answer

Using the Distributive Property, 2x + 3x can be written as (2 + 3)x, which simplifies to 5x.

Step-by-step solution

- Understand the Distributive Property

The Distributive Property states that for any numbers or variables, the expression a(b + c) equals ab + ac. This means a number or variable multiplying a sum can be distributed to multiply each term within the sum individually.

- Write the Expression using Distribution

Starting with the expression 2x + 3x, imagine grouping the terms under a common factor. Here, the common factor is x. Rewriting, it looks like: 2x + 3x = (2 + 3)x

- Perform the Addition

Add the coefficients (the numbers in front of x): (2 + 3)x = 5x

- Conclude the Justification

Using the Distributive Property, it has been shown that: 2x + 3x = (2 + 3)x = 5x. Therefore, 2x + 3x simplifies to 5x.

Key concepts

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An algebraic expression is a combination of numbers, variables (like x or y), and arithmetic operations (such as +, -, *, and /). In the given example, the expression is 2x + 3x. The variables in this expression are both 'x', and the coefficients are 2 and 3, respectively. Understanding how these components work together is the foundation of mastering algebra.

The Distributive Property comes into play here. This property allows us to simplify and solve expressions by distributing a common factor across the terms inside parentheses. Imagine the expression 2x + 3x as two groups containing 'x'. We group these terms under a common factor, which, in this case, is 'x'. When we rewrite the expression as (2 + 3)x, we effectively use the Distributive Property.

By performing this distribution step, the expression 2x + 3x becomes (2 + 3)x, which is much simpler to solve.

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Combining like terms is a crucial step in simplifying algebraic expressions. Like terms are terms whose variables and their exponents are the same. In the expression 2x + 3x, both terms are like terms because they both have the variable 'x' with the same exponent.

To combine these like terms, you focus on the coefficients – the numbers in front of the variables. The coefficients in 2x + 3x are 2 and 3. By adding these coefficients, you get 2 + 3, which equals 5.

This addition step simplifies the expression from 2x + 3x to 5x. When you combine like terms, you make the expression easier to work with and solve, just like grouping items to count them more efficiently.

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Mathematical justification involves explaining why a certain solution or simplification process is valid using established mathematical principles. Here, using the Distributive Property for the given expression serves as our justification.

To justify that 2x + 3x equals 5x, we break down the problem using the Distributive Property. By rewriting 2x + 3x as (2 + 3)x, we distribute the 'x' to both the 2 and the 3, effectively combining the coefficients under a common factor. This step demonstrates why adding the coefficients of like terms works.

Furthermore, by following through the steps: rewriting 2x + 3x as (2 + 3)x, performing the addition to get 5x, and concluding with 2x + 3x = 5x, we use a clear, logical process to justify our solution. Mathematical justifications help ensure that our solutions are not only correct but also understandable and consistent with mathematical principles.