Which Procedure Justifies Whether $-3x(5-4)+3(x-6)$ Is Equivalent To $-12x-6$?A. The Expressions Are Not Equivalent Because $-3(2)(5-4)+3(2-6)=-18$ And $ − 12 ( 2 ) − 6 = − 30 -12(2)-6=-30 − 12 ( 2 ) − 6 = − 30 [/tex].B. The Expressions Are Not
Understanding the Problem
When evaluating the equivalence of algebraic expressions, it's essential to follow a systematic approach to ensure accuracy. In this article, we will delve into the world of algebra and explore the procedure for determining whether two given expressions are equivalent. We will examine the expressions $-3x(5-4)+3(x-6)$ and $-12x-6$ to determine if they are equivalent.
The Importance of Following the Order of Operations
To begin, let's recall the order of operations, which is a set of rules that dictate the order in which mathematical operations should be performed. The acronym PEMDAS is commonly used to remember the order of operations:
- Parentheses: Evaluate expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
Evaluating the First Expression
Let's begin by evaluating the first expression, $-3x(5-4)+3(x-6)$. To do this, we need to follow the order of operations.
Step 1: Evaluate the Expression Inside the Parentheses
The first step is to evaluate the expression inside the parentheses: $(5-4)$. This simplifies to $1$.
Step 2: Multiply $-3x$ by the Result
Next, we multiply $-3x$ by the result: $-3x(1)$. This simplifies to $-3x$.
Step 3: Evaluate the Expression Inside the Second Set of Parentheses
Now, let's evaluate the expression inside the second set of parentheses: $(x-6)$. This simplifies to $x-6$.
Step 4: Multiply $3$ by the Result
Next, we multiply $3$ by the result: $3(x-6)$. This simplifies to $3x-18$.
Step 5: Combine the Results
Finally, we combine the results: $-3x+3x-18$. This simplifies to $-18$.
Evaluating the Second Expression
Now, let's evaluate the second expression, $-12x-6$. This expression is already simplified, so we can move on to the next step.
Comparing the Results
Now that we have evaluated both expressions, we can compare the results. The first expression simplifies to $-18$, while the second expression simplifies to $-12x-6$. To determine if these expressions are equivalent, we need to consider the values of $x$.
The Final Answer
The final answer is: A
Explanation
The expressions are not equivalent because the first expression simplifies to a constant value, $-18$, while the second expression simplifies to an expression involving the variable $x$, $-12x-6$. This means that the two expressions are not equivalent for all values of $x$.
Conclusion
In conclusion, evaluating the equivalence of algebraic expressions requires a systematic approach. By following the order of operations and simplifying the expressions, we can determine if two expressions are equivalent. In this case, the expressions $-3x(5-4)+3(x-6)$ and $-12x-6$ are not equivalent because the first expression simplifies to a constant value, while the second expression simplifies to an expression involving the variable $x$.
Q: What is the order of operations?
A: The order of operations is a set of rules that dictate the order in which mathematical operations should be performed. The acronym PEMDAS is commonly used to remember the order of operations:
- Parentheses: Evaluate expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
Q: How do I evaluate an expression with multiple operations?
A: To evaluate an expression with multiple operations, follow the order of operations. Start by evaluating any expressions inside parentheses, then evaluate any exponential expressions, followed by any multiplication and division operations, and finally any addition and subtraction operations.
Q: What is the difference between equivalent and non-equivalent expressions?
A: Equivalent expressions are expressions that have the same value for all possible values of the variables. Non-equivalent expressions are expressions that do not have the same value for all possible values of the variables.
Q: How do I determine if two expressions are equivalent?
A: To determine if two expressions are equivalent, simplify both expressions and compare the results. If the simplified expressions are the same, then the original expressions are equivalent.
Q: What is the importance of following the order of operations?
A: Following the order of operations is crucial when evaluating expressions with multiple operations. If the order of operations is not followed, the result may be incorrect.
Q: Can two expressions be equivalent even if they look different?
A: Yes, two expressions can be equivalent even if they look different. For example, the expressions $2x+3$ and $x+2x+3$ are equivalent, even though they look different.
Q: How do I simplify an expression with variables?
A: To simplify an expression with variables, combine like terms and follow the order of operations.
Q: What is the difference between a variable and a constant?
A: A variable is a symbol that represents a value that can change, while a constant is a value that does not change.
Q: Can a variable be a constant?
A: No, a variable cannot be a constant. A variable is a symbol that represents a value that can change, while a constant is a value that does not change.
Q: How do I evaluate an expression with multiple variables?
A: To evaluate an expression with multiple variables, substitute the values of the variables into the expression and follow the order of operations.
Q: What is the importance of evaluating expressions with multiple variables?
A: Evaluating expressions with multiple variables is crucial in many areas of mathematics and science, such as algebra, calculus, and physics.
Q: Can two expressions be equivalent even if they have different variables?
A: Yes, two expressions can be equivalent even if they have different variables. For example, the expressions $2x+3$ and $2y+3$ are equivalent, even though they have different variables.
Q: How do I determine if two expressions are equivalent when they have different variables?
A: To determine if two expressions are equivalent when they have different variables, substitute the values of the variables into the expressions and compare the results.
Q: What is the difference between an expression and an equation?
A: An expression is a mathematical statement that contains variables and constants, while an equation is a mathematical statement that contains an equal sign (=) and is used to solve for a variable.
Q: Can an expression be an equation?
A: Yes, an expression can be an equation. For example, the expression $2x+3=5$ is an equation.
Q: How do I solve an equation with an expression?
A: To solve an equation with an expression, isolate the variable by following the order of operations and using inverse operations.
Q: What is the importance of solving equations with expressions?
A: Solving equations with expressions is crucial in many areas of mathematics and science, such as algebra, calculus, and physics.
Q: Can two expressions be equivalent even if they have different equations?
A: Yes, two expressions can be equivalent even if they have different equations. For example, the expressions $2x+3$ and $x+2x+3$ are equivalent, even though they have different equations.
Q: How do I determine if two expressions are equivalent when they have different equations?
A: To determine if two expressions are equivalent when they have different equations, substitute the values of the variables into the expressions and compare the results.
Q: What is the difference between an expression and a function?
A: An expression is a mathematical statement that contains variables and constants, while a function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range).
Q: Can an expression be a function?
A: Yes, an expression can be a function. For example, the expression $f(x)=2x+3$ is a function.
Q: How do I determine if an expression is a function?
A: To determine if an expression is a function, check if the expression has a single output for each input.
Q: What is the importance of determining if an expression is a function?
A: Determining if an expression is a function is crucial in many areas of mathematics and science, such as algebra, calculus, and physics.
Q: Can two expressions be equivalent even if they have different functions?
A: Yes, two expressions can be equivalent even if they have different functions. For example, the expressions $2x+3$ and $x+2x+3$ are equivalent, even though they have different functions.
Q: How do I determine if two expressions are equivalent when they have different functions?
A: To determine if two expressions are equivalent when they have different functions, substitute the values of the variables into the expressions and compare the results.
Q: What is the difference between an expression and a relation?
A: An expression is a mathematical statement that contains variables and constants, while a relation is a set of ordered pairs that satisfy a certain condition.
Q: Can an expression be a relation?
A: Yes, an expression can be a relation. For example, the expression $y=2x+3$ is a relation.
Q: How do I determine if an expression is a relation?
A: To determine if an expression is a relation, check if the expression has a set of ordered pairs that satisfy a certain condition.
Q: What is the importance of determining if an expression is a relation?
A: Determining if an expression is a relation is crucial in many areas of mathematics and science, such as algebra, calculus, and physics.
Q: Can two expressions be equivalent even if they have different relations?
A: Yes, two expressions can be equivalent even if they have different relations. For example, the expressions $2x+3$ and $x+2x+3$ are equivalent, even though they have different relations.
Q: How do I determine if two expressions are equivalent when they have different relations?
A: To determine if two expressions are equivalent when they have different relations, substitute the values of the variables into the expressions and compare the results.