James Determined That These Two Expressions Were Equivalent Using The Values Of { X = 4 $}$ And { X = 6 $}$.Given Expressions: ${ 7x + 4 }$ ${ 3x + 5 + 4x - 1 }$Which Statements Are True? Check All That

Introduction

In mathematics, algebraic expressions are used to represent mathematical relationships between variables and constants. Two algebraic expressions are considered equivalent if they have the same value for a given set of values of the variables. In this article, we will explore the equivalence of two given algebraic expressions using the values of $x = 4$ and $x = 6$. We will also examine the statements that are true regarding the equivalence of these expressions.

The Given Expressions

The two given algebraic expressions are:

1. $7x + 4$
2. $3x + 5 + 4x - 1$

Evaluating the Expressions

To determine if the two expressions are equivalent, we need to evaluate them for the given values of $x$. Let's start by substituting $x = 4$ into both expressions.

Expression 1: $7x + 4$

Substituting $x = 4$ into the first expression, we get:

$7(4) + 4 = 28 + 4 = 32$

Expression 2: $3x + 5 + 4x - 1$

Substituting $x = 4$ into the second expression, we get:

$3(4) + 5 + 4(4) - 1 = 12 + 5 + 16 - 1 = 32$

As we can see, both expressions evaluate to the same value, $32$, when $x = 4$.

Evaluating the Expressions for $x = 6$

Now, let's substitute $x = 6$ into both expressions.

Expression 1: $7x + 4$

Substituting $x = 6$ into the first expression, we get:

$7(6) + 4 = 42 + 4 = 46$

Expression 2: $3x + 5 + 4x - 1$

Substituting $x = 6$ into the second expression, we get:

$3(6) + 5 + 4(6) - 1 = 18 + 5 + 24 - 1 = 46$

Again, both expressions evaluate to the same value, $46$, when $x = 6$.

Analyzing the Results

Based on the evaluations, we can see that both expressions are equivalent for the given values of $x$. However, we need to examine the statements that are true regarding the equivalence of these expressions.

Statement 1: The expressions are equivalent for all values of $x$.

This statement is false. The expressions are equivalent only for the specific values of $x$ that we evaluated, $x = 4$ and $x = 6$.

Statement 2: The expressions are equivalent for $x = 4$.

This statement is true. We evaluated both expressions for $x = 4$ and found that they have the same value.

Statement 3: The expressions are equivalent for $x = 6$.

This statement is true. We evaluated both expressions for $x = 6$ and found that they have the same value.

Statement 4: The expressions are equivalent for all values of $x$ except $x = 4$.

This statement is false. We found that the expressions are equivalent for both $x = 4$ and $x = 6$.

Statement 5: The expressions are equivalent for all values of $x$ except $x = 6$.

This statement is false. We found that the expressions are equivalent for both $x = 4$ and $x = 6$.

Conclusion

In conclusion, the two given algebraic expressions are equivalent for the specific values of $x$ that we evaluated, $x = 4$ and $x = 6$. However, we cannot conclude that the expressions are equivalent for all values of $x$. The statements that are true regarding the equivalence of these expressions are:

• The expressions are equivalent for $x = 4$.
• The expressions are equivalent for $x = 6$.

Recommendations

When working with algebraic expressions, it is essential to evaluate them for specific values of the variables to determine their equivalence. In this case, we found that the expressions are equivalent for the specific values of $x$ that we evaluated, but not for all values of $x$. Therefore, we should be cautious when making general statements about the equivalence of algebraic expressions.

Final Thoughts

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Introduction

In our previous article, we explored the equivalence of two given algebraic expressions using the values of $x = 4$ and $x = 6$. We found that the expressions are equivalent for the specific values of $x$ that we evaluated, but not for all values of $x$. In this article, we will answer some frequently asked questions (FAQs) about evaluating the equivalence of algebraic expressions.

Q: What is the purpose of evaluating the equivalence of algebraic expressions?

A: The purpose of evaluating the equivalence of algebraic expressions is to determine if two or more expressions have the same value for a given set of values of the variables. This is essential in mathematics, as it helps us to simplify complex expressions, identify equivalent expressions, and solve equations.

Q: How do I evaluate the equivalence of algebraic expressions?

A: To evaluate the equivalence of algebraic expressions, you need to substitute the values of the variables into the expressions and simplify them. If the expressions have the same value for a given set of values of the variables, then they are equivalent.

Q: What are some common mistakes to avoid when evaluating the equivalence of algebraic expressions?

A: Some common mistakes to avoid when evaluating the equivalence of algebraic expressions include:

• Not simplifying the expressions before comparing them
• Not using the correct values of the variables
• Not considering all possible values of the variables
• Not checking for equivalent expressions that may have different forms

Q: Can two algebraic expressions be equivalent even if they have different forms?

A: Yes, two algebraic expressions can be equivalent even if they have different forms. For example, the expressions $x^2 + 4x + 4$ and $(x + 2)^2$ are equivalent, even though they have different forms.

Q: How do I determine if two algebraic expressions are equivalent?

A: To determine if two algebraic expressions are equivalent, you need to follow these steps:

1. Simplify both expressions
2. Substitute the values of the variables into both expressions
3. Compare the values of both expressions
4. If the expressions have the same value for a given set of values of the variables, then they are equivalent

Q: Can two algebraic expressions be equivalent for some values of the variables but not for others?

A: Yes, two algebraic expressions can be equivalent for some values of the variables but not for others. For example, the expressions $x^2 + 4x + 4$ and $(x + 2)^2$ are equivalent for all values of $x$, but the expressions $x^2 + 4x + 4$ and $x^2 + 5x + 6$ are equivalent only for some values of $x$.

Q: How do I use algebraic expressions to solve equations?

A: Algebraic expressions can be used to solve equations by substituting the values of the variables into the expressions and simplifying them. If the expressions have the same value for a given set of values of the variables, then the equation is true for those values of the variables.

Q: Can I use algebraic expressions to solve systems of equations?

A: Yes, algebraic expressions can be used to solve systems of equations by substituting the values of the variables into the expressions and simplifying them. If the expressions have the same value for a given set of values of the variables, then the system of equations is true for those values of the variables.

Conclusion

In conclusion, evaluating the equivalence of algebraic expressions is an essential skill in mathematics. By following the steps outlined in this article, you can determine if two or more expressions have the same value for a given set of values of the variables. Remember to simplify the expressions, substitute the values of the variables, and compare the values of the expressions to determine if they are equivalent.