Simplify Algebraic Expressions Using The Distributive Property

Let's dive into simplifying algebraic expressions, a fundamental skill in mathematics. Today, we'll tackle the expression $3.1(x+1)+4(x-2.3)$ and use the distributive property to find its simplest form. You might have seen this kind of problem before, and the goal is always to make complex expressions more manageable. The distributive property is like a magic wand that helps us break down multiplication over addition or subtraction. It states that for any numbers a, b, and c, a(b+c) = ab + ac. We'll apply this rule to each part of our expression. So, grab your favorite pen and paper, and let's get started on unraveling this expression step-by-step. We'll cover the application of the distributive property, how to combine like terms, and finally arrive at the simplified answer, identifying the correct option among the choices provided. Understanding this process is key to mastering more advanced algebraic concepts, so let's ensure we build a solid foundation here.

Understanding the Distributive Property

The distributive property is a core concept in algebra that allows us to simplify expressions involving multiplication and addition or subtraction. In essence, it states that multiplying a sum by a number is the same as multiplying each addend by that number and then adding the products. Mathematically, this is represented as a(b + c) = ab + ac. Similarly, for subtraction, we have a(b - c) = ab - ac. This property is crucial because it helps us eliminate parentheses in algebraic expressions, making them easier to work with. When we encounter an expression like $3.1(x+1)$, the distributive property tells us to multiply $3.1$ by each term inside the parentheses. So, $3.1 imes x$ becomes $3.1x$, and $3.1 imes 1$ becomes $3.1$. Therefore, $3.1(x+1)$ expands to $3.1x + 3.1$. This step is fundamental because it transforms a product of a number and a binomial into a sum of two terms. We repeat this process for the second part of our expression, $4(x-2.3)$. Here, we multiply $4$ by $x$, which gives us $4x$, and then multiply $4$ by $-2.3$. It's important to remember the sign here; multiplying a positive number by a negative number results in a negative number. So, $4 imes -2.3 = -9.2$. Consequently, $4(x-2.3)$ expands to $4x - 9.2$. By applying the distributive property to both parts of the original expression, we have successfully removed the parentheses and are one step closer to finding the simplest form. This methodical approach ensures that we don't miss any terms or make sign errors, which are common pitfalls when first learning these concepts. The distributive property isn't just a rule; it's a tool that simplifies complexity and lays the groundwork for solving equations and manipulating more intricate algebraic structures.

Applying the Distributive Property to the Expression

Now that we understand the distributive property, let's apply it directly to our given expression: $3.1(x+1)+4(x-2.3)$. Our first step is to distribute the $3.1$ into the first set of parentheses $(x+1)$. As we discussed, this means multiplying $3.1$ by both $x$ and $1$. This gives us $3.1 imes x + 3.1 imes 1$, which simplifies to $3.1x + 3.1$. This part of the expression is now expanded. Next, we turn our attention to the second part of the expression: $4(x-2.3)$. Here, we distribute the $4$ into the parentheses $(x-2.3)$. This involves multiplying $4$ by $x$ and then multiplying $4$ by $-2.3$. The multiplication of $4$ by $x$ yields $4x$. The multiplication of $4$ by $-2.3$ results in $-9.2$ (remember, a positive times a negative is a negative). So, the expanded form of $4(x-2.3)$ is $4x - 9.2$. Now, we combine the results of these two distributive steps. Our original expression $3.1(x+1)+4(x-2.3)$ becomes $3.1x + 3.1 + 4x - 9.2$. At this stage, we have successfully removed all the parentheses. This is a significant achievement in simplifying the expression. The next crucial step will be to combine the like terms. This process of distribution is fundamental in algebra, allowing us to transform expressions into a form where we can more easily solve for variables or perform further operations. By systematically applying the distributive property, we ensure that every term within the parentheses is accounted for, leading us closer to the final, simplified answer. This method prevents errors and builds confidence in tackling similar problems.

Combining Like Terms

After applying the distributive property, our expression is $3.1x + 3.1 + 4x - 9.2$. The next logical step in simplifying this expression is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our expression, the terms with the variable $x$ are $3.1x$ and $4x$. These are our like terms involving $x$. We also have constant terms, which are numbers without any variables. Our constant terms are $3.1$ and $-9.2$. To combine like terms, we simply add or subtract their coefficients (the numbers in front of the variables) or the constants themselves. First, let's combine the $x$ terms: $3.1x + 4x$. We add the coefficients: $3.1 + 4 = 7.1$. So, $3.1x + 4x$ becomes $7.1x$. Now, let's combine the constant terms: $3.1 - 9.2$. We perform the subtraction: $3.1 - 9.2 = -6.1$. So, $3.1 - 9.2$ becomes $-6.1$. Once we have combined both sets of like terms, we put them back together to form the simplified expression. The combined $x$ terms give us $7.1x$, and the combined constant terms give us $-6.1$. Therefore, the simplest form of the expression is $7.1x - 6.1$. This process of combining like terms is essential for reducing an expression to its most concise form, making it easier to analyze and use in further calculations. It’s like sorting items into different boxes – all the $x$'s go together, and all the plain numbers go together. This systematic grouping and combining ensure that the final expression accurately represents the original one, just in a much neater package.

Identifying the Correct Option

We have successfully simplified the expression $3.1(x+1)+4(x-2.3)$ step-by-step. First, we used the distributive property to expand the expression, resulting in $3.1x + 3.1 + 4x - 9.2$. Then, we combined the like terms. We combined the $x$ terms ($3.1x + 4x$) to get $7.1x$, and we combined the constant terms ($3.1 - 9.2$) to get $-6.1$. Putting these together, the simplest form of the expression is $7.1x - 6.1$. Now, let's compare this result with the given options:

• A. $3.1 x+3.1+4 x-9.2$: This option represents the expression after applying the distributive property but before combining like terms. It is not the simplest form.
• B. $0.9 x+9.2$: This option has incorrect coefficients and constants, indicating errors in the application of the distributive property or in combining terms.
• C. $7.1 x-6.1$: This option exactly matches our simplified expression. It correctly combines the $x$ terms and the constant terms.
• D. $12.4 x+6.1$: This option appears to be the result of incorrectly adding coefficients ($3.1+4=7.1$, not $12.4$) and mishandling the constant terms.

Therefore, the correct option that represents the expression in its simplest form is C. $7.1 x-6.1$. This conclusion validates our step-by-step simplification process, demonstrating the power of applying the distributive property and combining like terms accurately.

Conclusion

We've successfully navigated the process of simplifying the algebraic expression $3.1(x+1)+4(x-2.3)$ using the fundamental distributive property. By distributing the coefficients $3.1$ and $4$ into their respective parentheses, we transformed the expression into $3.1x + 3.1 + 4x - 9.2$. The subsequent crucial step involved combining like terms, where we grouped the $x$ terms ($3.1x$ and $4x$) and the constant terms ($3.1$ and $-9.2$). This consolidation led us to the simplified form of $7.1x - 6.1$. We confirmed that Option C accurately reflects this result. Mastering these techniques is vital for proficiency in algebra, enabling you to tackle more complex equations and problems with confidence. The distributive property and the ability to combine like terms are foundational tools that appear throughout your mathematical journey. Keep practicing these concepts, and you'll find that algebraic manipulation becomes second nature.

For further exploration and practice on algebraic expressions and the distributive property, you can visit Khan Academy, a fantastic resource for all things math!