Write An Equivalent Expression For: 1 3 ( 12 X + 6 X − 21 ) = □ X − □ \frac{1}{3}(12x + 6x - 21) = \square X - \square 3 1 ​ ( 12 X + 6 X − 21 ) = □ X − □

Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities. One of the most common techniques used to simplify expressions is to combine like terms. In this article, we will focus on simplifying a given expression and rewriting it in an equivalent form.

The Given Expression

The given expression is:

$\frac{1}{3}(12x + 6x - 21) = \square x - \square$

Our goal is to simplify the expression on the left-hand side and rewrite it in an equivalent form.

Step 1: Distribute the Fraction

To simplify the expression, we need to distribute the fraction $\frac{1}{3}$ to each term inside the parentheses.

$\frac{1}{3}(12x + 6x - 21) = \frac{1}{3}(12x) + \frac{1}{3}(6x) - \frac{1}{3}(21)$

Step 2: Simplify Each Term

Now, we can simplify each term by multiplying the fraction with the term.

$\frac{1}{3}(12x) = 4x$

$\frac{1}{3}(6x) = 2x$

$\frac{1}{3}(21) = 7$

Step 3: Combine Like Terms

The expression now becomes:

$4x + 2x - 7$

We can combine like terms by adding or subtracting the coefficients of the same variable.

$4x + 2x = 6x$

So, the simplified expression is:

$6x - 7$

Step 4: Rewrite the Expression in Equivalent Form

The simplified expression is:

$6x - 7$

We can rewrite this expression in an equivalent form by factoring out the greatest common factor (GCF) of the terms.

The GCF of $6x$ and $-7$ is $1$, but we can factor out a $1$ from each term.

$6x - 7 = 1(6x) - 1(7)$

However, this does not simplify the expression. We can try to factor out a common factor from the terms, but there is no common factor other than $1$.

Therefore, the simplified expression is:

$6x - 7$

Conclusion

In this article, we simplified a given expression and rewrote it in an equivalent form. We used the distributive property to distribute the fraction to each term inside the parentheses, and then simplified each term by multiplying the fraction with the term. We combined like terms by adding or subtracting the coefficients of the same variable, and finally, we rewrote the expression in an equivalent form by factoring out the greatest common factor.

Tips and Tricks

• When simplifying expressions, always look for like terms and combine them by adding or subtracting the coefficients of the same variable.
• Use the distributive property to distribute fractions to each term inside the parentheses.
• Simplify each term by multiplying the fraction with the term.
• Factor out the greatest common factor (GCF) of the terms to rewrite the expression in an equivalent form.

Practice Problems

1. Simplify the expression: $\frac{1}{2}(8x + 4x - 12) = \square x - \square$
2. Simplify the expression: $\frac{1}{4}(16x + 8x - 24) = \square x - \square$
3. Simplify the expression: $\frac{1}{3}(9x + 6x - 15) = \square x - \square$

Answer Key

1. $6x - 6$
2. $8x - 6$
3. $5x - 5$
   Simplifying Algebraic Expressions: A Q&A Guide =====================================================

Introduction

In our previous article, we simplified a given expression and rewrote it in an equivalent form. In this article, we will answer some frequently asked questions (FAQs) about simplifying algebraic expressions.

Q: What is the distributive property?

A: The distributive property is a mathematical concept that allows us to distribute a fraction or a coefficient to each term inside the parentheses. For example, if we have the expression $\frac{1}{2}(x + y)$, we can distribute the fraction to each term inside the parentheses as follows:

$\frac{1}{2}(x + y) = \frac{1}{2}x + \frac{1}{2}y$

Q: How do I simplify an expression with fractions?

A: To simplify an expression with fractions, we need to follow these steps:

1. Distribute the fraction to each term inside the parentheses.
2. Simplify each term by multiplying the fraction with the term.
3. Combine like terms by adding or subtracting the coefficients of the same variable.

For example, let's simplify the expression $\frac{1}{3}(12x + 6x - 21)$:

$\frac{1}{3}(12x + 6x - 21) = \frac{1}{3}(12x) + \frac{1}{3}(6x) - \frac{1}{3}(21)$

$= 4x + 2x - 7$

Q: What is the difference between like terms and unlike terms?

A: Like terms are terms that have the same variable and coefficient. Unlike terms are terms that have different variables or coefficients.

For example, $2x$ and $3x$ are like terms because they have the same variable ($x$) and coefficient (2 and 3, respectively). On the other hand, $2x$ and $3y$ are unlike terms because they have different variables ($x$ and $y$, respectively).

Q: How do I combine like terms?

A: To combine like terms, we need to add or subtract the coefficients of the same variable. For example, let's combine the like terms $2x$ and $3x$:

$2x + 3x = 5x$

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) is the largest factor that divides each term in an expression. For example, the GCF of $6x$ and $-7$ is $1$ because $1$ is the largest factor that divides both terms.

Q: How do I factor out the GCF?

A: To factor out the GCF, we need to divide each term in the expression by the GCF. For example, let's factor out the GCF from the expression $6x - 7$:

$6x - 7 = 1(6x) - 1(7)$

However, this does not simplify the expression. We can try to factor out a common factor from the terms, but there is no common factor other than $1$.

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

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

• Not distributing the fraction to each term inside the parentheses.
• Not simplifying each term by multiplying the fraction with the term.
• Not combining like terms by adding or subtracting the coefficients of the same variable.
• Not factoring out the greatest common factor (GCF) of the terms.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about simplifying algebraic expressions. We discussed the distributive property, simplifying expressions with fractions, like terms and unlike terms, combining like terms, the greatest common factor (GCF), and common mistakes to avoid when simplifying expressions.

Tips and Tricks

• Always distribute the fraction to each term inside the parentheses.
• Simplify each term by multiplying the fraction with the term.
• Combine like terms by adding or subtracting the coefficients of the same variable.
• Factor out the greatest common factor (GCF) of the terms.
• Avoid common mistakes such as not distributing the fraction, not simplifying each term, and not combining like terms.

Practice Problems

1. Simplify the expression: $\frac{1}{2}(8x + 4x - 12) = \square x - \square$
2. Simplify the expression: $\frac{1}{4}(16x + 8x - 24) = \square x - \square$
3. Simplify the expression: $\frac{1}{3}(9x + 6x - 15) = \square x - \square$

Answer Key

1. $6x - 6$
2. $8x - 6$
3. $5x - 5$