Simplify The Expression:$\[ 6x + 7 + X - 8 \\]

Understanding the Problem

In this problem, we are given an algebraic expression that needs to be simplified. The expression is: $6x + 7 + x - 8$. Our goal is to combine like terms and simplify the expression to its simplest form.

Step 1: Identify Like Terms

To simplify the expression, we need to identify like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have two like terms: $6x$ and $x$. We also have two constant terms: $7$ and $-8$.

Step 2: Combine Like Terms

Now that we have identified like terms, we can combine them. To combine like terms, we add or subtract their coefficients. In this case, we have:

• $6x + x = 7x$ (combining like terms with the same variable)
• $7 + (-8) = -1$ (combining constant terms)

So, the simplified expression is: $7x - 1$.

Step 3: Check the Simplified Expression

To check the simplified expression, we can plug in a value for $x$ and see if the original expression and the simplified expression are equal. Let's say we plug in $x = 2$. Then, the original expression becomes:

$6(2) + 7 + 2 - 8 = 12 + 7 + 2 - 8 = 13$

And the simplified expression becomes:

$7(2) - 1 = 14 - 1 = 13$

As we can see, the original expression and the simplified expression are equal, which confirms that our simplified expression is correct.

Conclusion

In this problem, we simplified the expression $6x + 7 + x - 8$ by combining like terms. We identified like terms, combined them, and checked the simplified expression to make sure it was correct. The simplified expression is $7x - 1$.

Real-World Applications

Simplifying expressions is an important skill in mathematics and has many real-world applications. For example, in physics, we often need to simplify complex expressions to solve problems. In engineering, we use algebraic expressions to design and optimize systems. In economics, we use algebraic expressions to model and analyze economic systems.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions:

• Combine like terms first: When simplifying an expression, combine like terms first. This will make it easier to simplify the expression.
• Use the distributive property: The distributive property states that $a(b + c) = ab + ac$. We can use this property to simplify expressions by distributing the terms.
• Check your work: Always check your work by plugging in a value for the variable and seeing if the original expression and the simplified expression are equal.

Common Mistakes

Here are some common mistakes to avoid when simplifying expressions:

• Not combining like terms: Failing to combine like terms can lead to incorrect simplified expressions.
• Not checking your work: Failing to check your work can lead to incorrect simplified expressions.
• Using the wrong order of operations: Using the wrong order of operations can lead to incorrect simplified expressions.

Simplifying Expressions with Variables

Simplifying expressions with variables is similar to simplifying expressions with constants. We need to combine like terms and use the distributive property to simplify the expression.

Example 1: Simplifying an Expression with Variables

Simplify the expression: $3x + 2x - 4$

Step 1: Identify Like Terms

In this case, we have two like terms: $3x$ and $2x$. We also have a constant term: $-4$.

Step 2: Combine Like Terms

Now that we have identified like terms, we can combine them. To combine like terms, we add or subtract their coefficients. In this case, we have:

• $3x + 2x = 5x$ (combining like terms with the same variable)
• $-4$ (constant term remains the same)

So, the simplified expression is: $5x - 4$.

Example 2: Simplifying an Expression with Variables

Simplify the expression: $2x + 3x + 2$

Step 1: Identify Like Terms

In this case, we have two like terms: $2x$ and $3x$. We also have a constant term: $2$.

Step 2: Combine Like Terms

Now that we have identified like terms, we can combine them. To combine like terms, we add or subtract their coefficients. In this case, we have:

• $2x + 3x = 5x$ (combining like terms with the same variable)
• $2$ (constant term remains the same)

So, the simplified expression is: $5x + 2$.

Conclusion

Q: What is the first step in simplifying an expression?

A: The first step in simplifying an expression is to identify like terms. Like terms are terms that have the same variable raised to the same power.

Q: How do I identify like terms?

A: To identify like terms, look for terms that have the same variable and the same exponent. For example, $2x$ and $3x$ are like terms because they both have the variable $x$ raised to the power of 1.

Q: What is the next step after identifying like terms?

A: After identifying like terms, combine them by adding or subtracting their coefficients. For example, $2x + 3x = 5x$.

Q: What is the distributive property?

A: The distributive property is a rule that states that $a(b + c) = ab + ac$. We can use this property to simplify expressions by distributing the terms.

Q: How do I use the distributive property to simplify an expression?

A: To use the distributive property to simplify an expression, multiply each term inside the parentheses by the term outside the parentheses. For example, $2(x + 3) = 2x + 6$.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when simplifying an expression. The order of operations is:

1. Parentheses
2. Exponents
3. Multiplication and Division
4. Addition and Subtraction

Q: Why is it important to follow the order of operations?

A: Following the order of operations is important because it ensures that we simplify expressions correctly. If we don't follow the order of operations, we may get incorrect answers.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

• Not combining like terms
• Not checking your work
• Using the wrong order of operations

Q: How do I check my work when simplifying an expression?

A: To check your work when simplifying an expression, plug in a value for the variable and see if the original expression and the simplified expression are equal.

Q: What are some real-world applications of simplifying expressions?

A: Simplifying expressions has many real-world applications, including:

• Physics: Simplifying expressions is used to solve problems in physics, such as calculating the trajectory of a projectile.
• Engineering: Simplifying expressions is used to design and optimize systems, such as bridges and buildings.
• Economics: Simplifying expressions is used to model and analyze economic systems, such as supply and demand.

Q: What are some tips and tricks for simplifying expressions?

A: Some tips and tricks for simplifying expressions include:

• Combine like terms first
• Use the distributive property
• Check your work
• Use the order of operations

Q: What are some common expressions that can be simplified?

A: Some common expressions that can be simplified include:

• $2x + 3x$
• $x + 2x$
• $3x - 2x$
• $2x + 5 - 3x$

Q: How do I simplify an expression with variables and constants?

A: To simplify an expression with variables and constants, combine like terms and use the distributive property. For example, $2x + 3 + 2x - 4 = 4x - 1$.

Q: What are some examples of expressions that can be simplified using the distributive property?

A: Some examples of expressions that can be simplified using the distributive property include:

• $2(x + 3) = 2x + 6$
• $3(x - 2) = 3x - 6$
• $4(x + 1) = 4x + 4$

Conclusion

In this article, we answered some common questions about simplifying expressions, including how to identify like terms, how to use the distributive property, and how to check your work. We also discussed real-world applications, tips and tricks, and common mistakes to avoid. We hope this article has helped you understand how to simplify expressions.