18. Select All Expressions Equivalent To $\frac{3}{5} X + 3$.A. $\frac{2}{5} X + 3 + \frac{1}{5} X$B. $\frac{4}{5} X - \frac{1}{5} X + 3$C. $\frac{2}{5} X + 3 + \frac{3}{5} X - 1$D. $1 + \frac{3}{5} X + 2$E.

18. Select all expressions equivalent to $\frac{3}{5} x + 3$

Understanding Equivalent Expressions in Algebra

In algebra, equivalent expressions are those that have the same value for all possible values of the variables involved. In this problem, we are given an expression $\frac{3}{5} x + 3$ and asked to select all the expressions that are equivalent to it. To solve this problem, we need to understand the concept of equivalent expressions and how to simplify algebraic expressions.

What are Equivalent Expressions?

Equivalent expressions are expressions that have the same value for all possible values of the variables involved. In other words, if two expressions are equivalent, they will always produce the same result when evaluated for any given value of the variables. For example, the expressions $2x + 3$ and $x + 5$ are equivalent because they both represent the same linear function.

Simplifying Algebraic Expressions

To simplify an algebraic expression, we need to combine like terms. Like terms are terms that have the same variable raised to the same power. For example, the terms $2x$ and $x$ are like terms because they both have the variable $x$ raised to the power of 1. We can combine like terms by adding or subtracting their coefficients.

Evaluating the Options

Now, let's evaluate the options given in the problem.

A. $\frac{2}{5} x + 3 + \frac{1}{5} x$

To determine if this expression is equivalent to $\frac{3}{5} x + 3$, we need to simplify it by combining like terms.

$\frac{2}{5} x + 3 + \frac{1}{5} x = \frac{2}{5} x + \frac{1}{5} x + 3$

Combining like terms, we get:

$\frac{2}{5} x + \frac{1}{5} x = \frac{3}{5} x$

So, the expression $\frac{2}{5} x + 3 + \frac{1}{5} x$ is equivalent to $\frac{3}{5} x + 3$.

B. $\frac{4}{5} x - \frac{1}{5} x + 3$

To determine if this expression is equivalent to $\frac{3}{5} x + 3$, we need to simplify it by combining like terms.

$\frac{4}{5} x - \frac{1}{5} x + 3 = \frac{4}{5} x - \frac{1}{5} x + \frac{15}{5}$

Combining like terms, we get:

$\frac{4}{5} x - \frac{1}{5} x = \frac{3}{5} x$

So, the expression $\frac{4}{5} x - \frac{1}{5} x + 3$ is equivalent to $\frac{3}{5} x + 3$.

C. $\frac{2}{5} x + 3 + \frac{3}{5} x - 1$

To determine if this expression is equivalent to $\frac{3}{5} x + 3$, we need to simplify it by combining like terms.

$\frac{2}{5} x + 3 + \frac{3}{5} x - 1 = \frac{2}{5} x + \frac{3}{5} x + 2$

Combining like terms, we get:

$\frac{2}{5} x + \frac{3}{5} x = \frac{5}{5} x = x$

So, the expression $\frac{2}{5} x + 3 + \frac{3}{5} x - 1$ is not equivalent to $\frac{3}{5} x + 3$.

D. $1 + \frac{3}{5} x + 2$

To determine if this expression is equivalent to $\frac{3}{5} x + 3$, we need to simplify it by combining like terms.

$1 + \frac{3}{5} x + 2 = \frac{5}{5} + \frac{3}{5} x + \frac{10}{5}$

Combining like terms, we get:

$\frac{5}{5} + \frac{10}{5} = \frac{15}{5} = 3$

So, the expression $1 + \frac{3}{5} x + 2$ is equivalent to $\frac{3}{5} x + 3$.

E.

There is no expression given for option E.

Conclusion

In conclusion, the expressions $\frac{2}{5} x + 3 + \frac{1}{5} x$, $\frac{4}{5} x - \frac{1}{5} x + 3$, and $1 + \frac{3}{5} x + 2$ are equivalent to $\frac{3}{5} x + 3$.
18. Select all expressions equivalent to $\frac{3}{5} x + 3$

Understanding Equivalent Expressions in Algebra

In algebra, equivalent expressions are those that have the same value for all possible values of the variables involved. In this problem, we are given an expression $\frac{3}{5} x + 3$ and asked to select all the expressions that are equivalent to it. To solve this problem, we need to understand the concept of equivalent expressions and how to simplify algebraic expressions.

What are Equivalent Expressions?

Equivalent expressions are expressions that have the same value for all possible values of the variables involved. In other words, if two expressions are equivalent, they will always produce the same result when evaluated for any given value of the variables. For example, the expressions $2x + 3$ and $x + 5$ are equivalent because they both represent the same linear function.

Simplifying Algebraic Expressions

To simplify an algebraic expression, we need to combine like terms. Like terms are terms that have the same variable raised to the same power. For example, the terms $2x$ and $x$ are like terms because they both have the variable $x$ raised to the power of 1. We can combine like terms by adding or subtracting their coefficients.

Evaluating the Options

Now, let's evaluate the options given in the problem.

A. $\frac{2}{5} x + 3 + \frac{1}{5} x$

To determine if this expression is equivalent to $\frac{3}{5} x + 3$, we need to simplify it by combining like terms.

$\frac{2}{5} x + 3 + \frac{1}{5} x = \frac{2}{5} x + \frac{1}{5} x + 3$

Combining like terms, we get:

$\frac{2}{5} x + \frac{1}{5} x = \frac{3}{5} x$

So, the expression $\frac{2}{5} x + 3 + \frac{1}{5} x$ is equivalent to $\frac{3}{5} x + 3$.

B. $\frac{4}{5} x - \frac{1}{5} x + 3$

To determine if this expression is equivalent to $\frac{3}{5} x + 3$, we need to simplify it by combining like terms.

$\frac{4}{5} x - \frac{1}{5} x + 3 = \frac{4}{5} x - \frac{1}{5} x + \frac{15}{5}$

Combining like terms, we get:

$\frac{4}{5} x - \frac{1}{5} x = \frac{3}{5} x$

So, the expression $\frac{4}{5} x - \frac{1}{5} x + 3$ is equivalent to $\frac{3}{5} x + 3$.

C. $\frac{2}{5} x + 3 + \frac{3}{5} x - 1$

To determine if this expression is equivalent to $\frac{3}{5} x + 3$, we need to simplify it by combining like terms.

$\frac{2}{5} x + 3 + \frac{3}{5} x - 1 = \frac{2}{5} x + \frac{3}{5} x + 2$

Combining like terms, we get:

$\frac{2}{5} x + \frac{3}{5} x = \frac{5}{5} x = x$

So, the expression $\frac{2}{5} x + 3 + \frac{3}{5} x - 1$ is not equivalent to $\frac{3}{5} x + 3$.

D. $1 + \frac{3}{5} x + 2$

To determine if this expression is equivalent to $\frac{3}{5} x + 3$, we need to simplify it by combining like terms.

$1 + \frac{3}{5} x + 2 = \frac{5}{5} + \frac{3}{5} x + \frac{10}{5}$

Combining like terms, we get:

$\frac{5}{5} + \frac{10}{5} = \frac{15}{5} = 3$

So, the expression $1 + \frac{3}{5} x + 2$ is equivalent to $\frac{3}{5} x + 3$.

E.

There is no expression given for option E.

Conclusion

In conclusion, the expressions $\frac{2}{5} x + 3 + \frac{1}{5} x$, $\frac{4}{5} x - \frac{1}{5} x + 3$, and $1 + \frac{3}{5} x + 2$ are equivalent to $\frac{3}{5} x + 3$.

Q&A

Q: What are equivalent expressions in algebra?

A: Equivalent expressions are expressions that have the same value for all possible values of the variables involved.

Q: How do you simplify algebraic expressions?

A: To simplify an algebraic expression, you need to combine like terms. Like terms are terms that have the same variable raised to the same power.

Q: What is the difference between equivalent expressions and similar expressions?

A: Equivalent expressions are expressions that have the same value for all possible values of the variables involved, while similar expressions are expressions that have the same form but may not have the same value.

Q: Can you give an example of equivalent expressions?

A: Yes, the expressions $2x + 3$ and $x + 5$ are equivalent because they both represent the same linear function.

Q: How do you determine if two expressions are equivalent?

A: To determine if two expressions are equivalent, you need to simplify them by combining like terms and then compare the resulting expressions.

Q: What is the importance of equivalent expressions in algebra?

A: Equivalent expressions are important in algebra because they allow us to simplify complex expressions and make it easier to solve equations and inequalities.

Q: Can you give an example of a real-world application of equivalent expressions?

A: Yes, equivalent expressions are used in many real-world applications, such as in physics and engineering, where they are used to simplify complex equations and make it easier to solve problems.

Q: How do you use equivalent expressions to solve equations and inequalities?

A: To use equivalent expressions to solve equations and inequalities, you need to simplify the expressions by combining like terms and then use the resulting expressions to solve the equation or inequality.

Q: What are some common mistakes to avoid when working with equivalent expressions?

A: Some common mistakes to avoid when working with equivalent expressions include not combining like terms, not simplifying the expressions, and not checking if the expressions are equivalent before using them to solve equations and inequalities.