Which Expression Is Equivalent To $\left(10+7r-r^2\right)+\left(-6r^2-18+5r\right$\]?
Introduction
Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the process of simplifying algebraic expressions, with a focus on combining like terms and applying the distributive property. We will also examine a specific example of an algebraic expression and demonstrate how to simplify it step by step.
What are Algebraic Expressions?
An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations. Variables are represented by letters, such as x or y, while constants are numbers that do not change value. Algebraic expressions can be simple, such as 2x + 3, or complex, involving multiple variables and operations.
The Importance of Simplifying Algebraic Expressions
Simplifying algebraic expressions is crucial in mathematics because it allows us to:
- Combine like terms: Simplifying expressions by combining like terms helps us to reduce the complexity of the expression and make it easier to work with.
- Apply the distributive property: The distributive property states that a single operation can be distributed over multiple terms. Simplifying expressions by applying the distributive property helps us to simplify complex expressions.
- Solve equations and inequalities: Simplifying algebraic expressions is essential for solving equations and inequalities. By simplifying expressions, we can isolate the variable and solve for its value.
Step-by-Step Guide to Simplifying Algebraic Expressions
Simplifying algebraic expressions involves several steps:
- Combine like terms: Combine terms that have the same variable and coefficient.
- Apply the distributive property: Distribute a single operation over multiple terms.
- Simplify constants: Simplify constants by combining them with other constants.
Example: Simplifying the Expression
Let's simplify the expression step by step.
Step 1: Combine Like Terms
The first step is to combine like terms. We can combine the terms with the same variable and coefficient.
import sympy as sp
# Define the variables
r = sp.symbols('r')
# Define the expression
expr = (10 + 7*r - r**2) + (-6*r**2 - 18 + 5*r)
# Simplify the expression
simplified_expr = sp.simplify(expr)
print(simplified_expr)
Step 2: Apply the Distributive Property
The next step is to apply the distributive property. We can distribute the addition operation over the terms.
# Define the expression
expr = (10 + 7*r - r**2) + (-6*r**2 - 18 + 5*r)
# Apply the distributive property
distributive_expr = sp.expand(expr)
print(distributive_expr)
Step 3: Simplify Constants
The final step is to simplify constants. We can combine the constants by adding or subtracting them.
# Define the expression
expr = (10 + 7*r - r**2) + (-6*r**2 - 18 + 5*r)
# Simplify constants
simplified_expr = sp.simplify(expr)
print(simplified_expr)
Conclusion
Q: What is the first step in simplifying an algebraic expression?
A: The first step in simplifying an algebraic expression is to combine like terms. Like terms are terms that have the same variable and coefficient.
Q: How do I combine like terms?
A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, if you have the expression 2x + 3x, you can combine the like terms by adding the coefficients: 2x + 3x = 5x.
Q: What is the distributive property?
A: The distributive property is a mathematical property that states that a single operation can be distributed over multiple terms. For example, if you have the expression 2(x + 3), you can apply the distributive property by multiplying 2 by each term inside the parentheses: 2x + 6.
Q: How do I apply the distributive property?
A: To apply the distributive property, you need to multiply the single operation by each term inside the parentheses. For example, if you have the expression 2(x + 3), you can apply the distributive property by multiplying 2 by each term inside the parentheses: 2x + 6.
Q: What is the difference between combining like terms and applying the distributive property?
A: Combining like terms involves adding or subtracting the coefficients of like terms, while applying the distributive property involves multiplying a single operation by each term inside the parentheses.
Q: How do I simplify constants?
A: To simplify constants, you need to add or subtract the constants. For example, if you have the expression 2 + 3, you can simplify the constants by adding them: 2 + 3 = 5.
Q: What is the final step in simplifying an algebraic expression?
A: The final step in simplifying an algebraic expression is to simplify constants. This involves adding or subtracting the constants to get the final simplified expression.
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 simplifying the expression manually to ensure that you get the correct answer.
Q: How do I know if an algebraic expression is simplified?
A: An algebraic expression is simplified when there are no like terms that can be combined, and the constants have been simplified. You can check if an expression is simplified by looking for any like terms that can be combined or any constants that can be simplified.
Q: Can I simplify an algebraic expression with multiple variables?
A: Yes, you can simplify an algebraic expression with multiple variables. The process of simplifying an algebraic expression with multiple variables is the same as simplifying an expression with a single variable.
Q: How do I simplify an algebraic expression with fractions?
A: To simplify an algebraic expression with fractions, you need to simplify the fractions by finding the least common denominator (LCD) and then combining the fractions.