Properties Used to Simplify the Expression

Oct 13, 2025 by recoveryshake.com 68 views

Understanding the Original Expression

The given original expression is ${$ (6x + 3) - (2x + 1)}$. This expression involves the combination of two algebraic expressions, and the goal is to simplify it step by step. To achieve this, we need to apply various mathematical properties to manipulate the expression and arrive at a simplified form.

Step 1: Applying the Distributive Property

The first step in simplifying the given expression is to apply the distributive property. This property states that for any real numbers a, b, and c, the following equation holds: a(b + c) = ab + ac. In the context of the given expression, we can apply the distributive property to expand the first term, (6x + 3), and then subtract the second term, (2x + 1).

Using the distributive property, we can rewrite the expression as:

${$ (6x + 3) - (2x + 1) = 6x + 3 - 2x - 1}$

Step 2: Combining Like Terms

The next step in simplifying the expression is to combine like terms. Like terms are terms that have the same variable raised to the same power. In the expression 6x + 3 - 2x - 1, we can combine the like terms 6x and -2x, as well as the constant terms 3 and -1.

Using the commutative and associative properties of addition, we can rearrange the terms to group like terms together:

${ $6x - 2x + 3 - 1}$

Step 3: Applying the Commutative and Associative Properties

The commutative property of addition states that the order of the terms does not change the result. The associative property of addition states that the order in which we add the terms does not change the result. We can use these properties to rearrange the terms and group like terms together.

Using the commutative and associative properties, we can rewrite the expression as:

${$ (6x - 2x) + (3 - 1)}$

Step 4: Simplifying the Expression

The final step in simplifying the expression is to simplify the terms. We can combine the like terms 6x and -2x, as well as the constant terms 3 and -1.

Using the distributive property, we can rewrite the expression as:

${ $4x + 2}$

Conclusion

In conclusion, we have simplified the given expression one step at a time by applying various mathematical properties. We started with the original expression ${$ (6x + 3) - (2x + 1)}$ and applied the distributive property, combined like terms, and used the commutative and associative properties to arrive at the simplified expression 4x + 2.

Properties Used to Simplify the Expression

The following properties were used to simplify the expression:

Distributive Property: a(b + c) = ab + ac
Commutative Property of Addition: a + b = b + a
Associative Property of Addition: (a + b) + c = a + (b + c)
Combining Like Terms: combining terms with the same variable raised to the same power

Importance of Mathematical Properties

Mathematical properties play a crucial role in simplifying algebraic expressions. By applying these properties, we can manipulate the expression and arrive at a simplified form. The distributive property, commutative property, associative property, and combining like terms are essential properties that help us simplify expressions and solve mathematical problems.

Real-World Applications

The concept of simplifying algebraic expressions has numerous real-world applications. In mathematics, algebraic expressions are used to model real-world problems, and simplifying these expressions helps us solve the problems. In physics, algebraic expressions are used to describe the motion of objects, and simplifying these expressions helps us understand the behavior of the objects. In engineering, algebraic expressions are used to design and optimize systems, and simplifying these expressions helps us improve the performance of the systems.

Conclusion

In conclusion, the given expression has been simplified one step at a time by applying various mathematical properties. We started with the original expression ${$ (6x + 3) - (2x + 1)}$ and applied the distributive property, combined like terms, and used the commutative and associative properties to arrive at the simplified expression 4x + 2. The properties used to simplify the expression are essential in mathematics and have numerous real-world applications.

Q: What is the distributive property, and how is it used to simplify algebraic expressions?

A: The distributive property is a mathematical property that states that for any real numbers a, b, and c, the following equation holds: a(b + c) = ab + ac. This property is used to simplify algebraic expressions by expanding the terms and combining like terms.

Q: What is the commutative property of addition, and how is it used to simplify algebraic expressions?

A: The commutative property of addition is a mathematical property that states that the order of the terms does not change the result. This property is used to simplify algebraic expressions by rearranging the terms and grouping like terms together.

Q: What is the associative property of addition, and how is it used to simplify algebraic expressions?

A: The associative property of addition is a mathematical property that states that the order in which we add the terms does not change the result. This property is used to simplify algebraic expressions by rearranging the terms and grouping like terms together.

Q: How do I combine like terms in an algebraic expression?

A: To combine like terms in an algebraic expression, you need to identify the terms that have the same variable raised to the same power. Then, you can add or subtract the coefficients of these terms to simplify the expression.

Q: What is the difference between a variable and a constant in an algebraic expression?

A: A variable is a letter or symbol that represents a value that can change, while a constant is a value that does not change. In an algebraic expression, variables are used to represent unknown values, while constants are used to represent known values.

Q: How do I simplify an algebraic expression with multiple variables?

A: To simplify an algebraic expression with multiple variables, you need to apply the distributive property, combine like terms, and use the commutative and associative properties to rearrange the terms and group like terms together.

Q: What is the importance of simplifying algebraic expressions?

A: Simplifying algebraic expressions is important because it helps us to:

Solve mathematical problems more easily
Understand the behavior of complex systems
Make predictions and decisions based on mathematical models
Improve the performance of systems and processes

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

A: To apply the distributive property to simplify an algebraic expression, you need to:

Identify the terms that need to be expanded
Multiply the coefficients of the terms by the variables
Combine like terms to simplify the expression

Q: What is the difference between an algebraic expression and an equation?

A: An algebraic expression is a mathematical statement that contains variables and constants, while an equation is a mathematical statement that contains an equal sign (=) and two expressions that are equal.

Q: How do I solve an algebraic equation?

A: To solve an algebraic equation, you need to:

Simplify the equation by combining like terms
Isolate the variable by adding, subtracting, multiplying, or dividing both sides of the equation
Check the solution by plugging it back into the original equation

Q: What is the importance of algebra in real-world applications?

A: Algebra is important in real-world applications because it helps us to:

Model complex systems and processes
Make predictions and decisions based on mathematical models
Improve the performance of systems and processes
Solve mathematical problems more easily

Q: How do I apply algebra to solve real-world problems?

A: To apply algebra to solve real-world problems, you need to:

Identify the variables and constants in the problem
Simplify the algebraic expression by combining like terms
Use the distributive property, commutative property, and associative property to rearrange the terms and group like terms together
Solve the equation by isolating the variable and checking the solution.