Select The Correct Answer.What Is The Simplified Form Of This Expression? { (4x - 4) - 8x$}$A. { -8x$}$B. { -4x - 4$}$C. { -4x + 4$}$D. ${ 12x - 4\$}
Understanding the Basics of Algebraic Simplification
Algebraic simplification is a crucial concept in mathematics that involves reducing complex expressions to their simplest form. This process helps in solving equations, inequalities, and other mathematical problems efficiently. In this article, we will focus on simplifying algebraic expressions, with a specific example to illustrate the concept.
The Expression to Simplify
The given expression is: {(4x - 4) - 8x$}$
Our goal is to simplify this expression by combining like terms and eliminating any unnecessary components.
Step 1: Distribute the Negative Sign
The first step in simplifying the expression is to distribute the negative sign to the terms inside the parentheses. This means multiplying each term inside the parentheses by -1.
{(4x - 4) - 8x = -4x + 4 - 8x$}$
Step 2: Combine Like Terms
Now that we have distributed the negative sign, we can combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have two terms with the variable x: -4x and -8x.
{-4x + 4 - 8x = -12x + 4$}$
Step 3: Simplify the Expression
The final step is to simplify the expression by eliminating any unnecessary components. In this case, we have a constant term (+4) that cannot be combined with any other terms.
{-12x + 4$}$
The Simplified Form
The simplified form of the expression is: {-12x + 4$}$
Comparing with the Options
Now that we have simplified the expression, let's compare it with the given options:
A. {-8x$}$ B. {-4x - 4$}$ C. {-4x + 4$}$ D. ${12x - 4\$}
Q: What is the purpose of simplifying algebraic expressions?
A: The purpose of simplifying algebraic expressions is to reduce complex expressions to their simplest form, making it easier to solve equations, inequalities, and other mathematical problems.
Q: What are like terms in algebraic expressions?
A: Like terms are terms that have the same variable raised to the same power. For example, 2x and 4x are like terms because they both have the variable x raised to the power of 1.
Q: How do I combine like terms in an algebraic expression?
A: To combine like terms, add or subtract the coefficients of the like terms. For example, 2x + 4x = 6x, and 3x - 2x = x.
Q: What is the difference between a coefficient and a variable?
A: A coefficient is a number that is multiplied by a variable. For example, in the expression 2x, 2 is the coefficient and x is the variable.
Q: Can I simplify an algebraic expression by adding or subtracting like terms?
A: Yes, you can simplify an algebraic expression by adding or subtracting like terms. This is a fundamental concept in algebra and is used to reduce complex expressions to their simplest form.
Q: How do I eliminate unnecessary components in an algebraic expression?
A: To eliminate unnecessary components in an algebraic expression, look for any terms that can be combined or eliminated. For example, if you have the expression 2x + 4, you can eliminate the constant term 4 by combining it with the variable term 2x.
Q: What is the final step in simplifying an algebraic expression?
A: The final step in simplifying an algebraic expression is to check your work and make sure that the expression is in its simplest form. This involves verifying that all like terms have been combined and that there are no unnecessary components.
Q: Can I use a calculator to simplify algebraic expressions?
A: Yes, you can use a calculator to simplify algebraic expressions. However, it's always a good idea to check your work by hand to make sure that the calculator is giving you the correct answer.
Q: What are some common mistakes to avoid when simplifying algebraic expressions?
A: Some common mistakes to avoid when simplifying algebraic expressions include:
- Forgetting to combine like terms
- Adding or subtracting unlike terms
- Eliminating necessary components
- Not checking your work
Q: How can I practice simplifying algebraic expressions?
A: You can practice simplifying algebraic expressions by working through examples and exercises in your textbook or online. You can also try simplifying expressions on your own and then checking your work with a calculator or a friend.
Q: What are some real-world applications of simplifying algebraic expressions?
A: Simplifying algebraic expressions has many real-world applications, including:
- Solving equations and inequalities in physics and engineering
- Modeling population growth and decline in biology
- Analyzing data in statistics and data analysis
- Solving optimization problems in economics and finance