Identifying the Step with the Distributive Property

Oct 13, 2025 by talynt.org 147 views

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that allows us to expand and simplify expressions by multiplying each term inside the parentheses with the term outside. In the context of solving equations, the distributive property is used to eliminate parentheses and simplify the equation. In this article, we will explore how to identify the step where Sarah used the distributive property in solving the equation $2(x+3)=8$ .

Step 1: Expanding the Equation

The first step in solving the equation is to expand the left-hand side using the distributive property. This involves multiplying each term inside the parentheses with the term outside. In this case, we multiply $2$ with each term inside the parentheses: $x$ and $3$ . This gives us the equation $2x+6=8$ .

2(x+3) = 2x + 2(3)

Step 2: Simplifying the Equation

In the second step, we simplify the equation by combining like terms. However, we do not use the distributive property in this step. Instead, we focus on isolating the variable $x$ by subtracting $6$ from both sides of the equation.

2x + 6 = 8
2x = 8 - 6
2x = 2

Step 3: Solving for x

In the third step, we solve for $x$ by dividing both sides of the equation by $2$ . This gives us the value of $x$ , which is $1$ .

2x = 2
x = 2/2
x = 1

Identifying the Step with the Distributive Property

Based on the steps outlined above, we can see that the distributive property is used in Step 1. This is where we expand the left-hand side of the equation by multiplying each term inside the parentheses with the term outside.

Conclusion

In conclusion, the distributive property is a powerful tool in algebra that allows us to expand and simplify expressions. By identifying the step where Sarah used the distributive property, we can better understand how to solve equations and simplify expressions. In this case, the distributive property is used in Step 1 to expand the left-hand side of the equation.

Answer

The correct answer is A. Step 1.

Additional Tips and Examples

The distributive property can be used to expand and simplify expressions with multiple terms inside the parentheses.
When using the distributive property, make sure to multiply each term inside the parentheses with the term outside.
The distributive property can be used to solve equations and simplify expressions.

Example 1: Expanding an Expression with Multiple Terms

Consider the expression $3(x+2y-4)$ . To expand this expression, we use the distributive property to multiply each term inside the parentheses with the term outside.

3(x+2y-4) = 3x + 3(2y) - 3(4)

Example 2: Solving an Equation with the Distributive Property

Consider the equation $4(x-2)=12$ . To solve this equation, we use the distributive property to expand the left-hand side.

4(x-2) = 4x - 4(2)
4x - 8 = 12
4x = 12 + 8
4x = 20
x = 20/4
x = 5

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to expand and simplify expressions by multiplying each term inside the parentheses with the term outside.

Q: When is the distributive property used?

A: The distributive property is used to expand and simplify expressions, solve equations, and simplify expressions with multiple terms inside the parentheses.

Q: How do I use the distributive property?

A: To use the distributive property, multiply each term inside the parentheses with the term outside. For example, if we have the expression $3(x+2y-4)$ , we would multiply $3$ with each term inside the parentheses: $x$ , $2y$ , and $-4$ .

Q: What is the difference between the distributive property and the commutative property?

A: The distributive property and the commutative property are two different concepts in algebra. The distributive property involves multiplying each term inside the parentheses with the term outside, while the commutative property involves rearranging the order of terms in an expression.

Q: Can I use the distributive property to simplify expressions with fractions?

A: Yes, you can use the distributive property to simplify expressions with fractions. For example, if we have the expression $\frac{1}{2}(x+2y-4)$ , we would multiply $\frac{1}{2}$ with each term inside the parentheses: $x$ , $2y$ , and $-4$ .

Q: How do I apply the distributive property to solve equations?

A: To apply the distributive property to solve equations, follow these steps:

Expand the left-hand side of the equation using the distributive property.
Simplify the equation by combining like terms.
Isolate the variable by adding, subtracting, multiplying, or dividing both sides of the equation.

Q: Can I use the distributive property to solve equations with multiple variables?

A: Yes, you can use the distributive property to solve equations with multiple variables. For example, if we have the equation $2(x+y+z)=12$ , we would expand the left-hand side using the distributive property and then simplify the equation.

Q: What are some common mistakes to avoid when using the distributive property?

A: Some common mistakes to avoid when using the distributive property include:

Forgetting to multiply each term inside the parentheses with the term outside.
Not simplifying the equation by combining like terms.
Not isolating the variable by adding, subtracting, multiplying, or dividing both sides of the equation.

Q: How can I practice using the distributive property?

A: You can practice using the distributive property by working through examples and exercises in your algebra textbook or online resources. You can also try creating your own examples and exercises to practice applying the distributive property.

Example 1: Expanding an Expression with Fractions

Consider the expression $\frac{1}{2}(x+2y-4)$ . To expand this expression, we use the distributive property to multiply each term inside the parentheses with the term outside.

\frac{1}{2}(x+2y-4) = \frac{1}{2}x + \frac{1}{2}(2y) - \frac{1}{2}(4)

Example 2: Solving an Equation with Multiple Variables

Consider the equation $2(x+y+z)=12$ . To solve this equation, we use the distributive property to expand the left-hand side and then simplify the equation.

2(x+y+z) = 2x + 2y + 2z
2x + 2y + 2z = 12

By following these examples and tips, you can become more comfortable using the distributive property to solve equations and simplify expressions.