Which Expression Is Equivalent To { (a+2)$}$ When { A=\beta$}$?A. ${ 7(6)\$} B. ${ 9(6)\$} C. ${ 7(\beta)+2\$} D. ${ 7(6)+14\$}

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Understanding the Problem

In this problem, we are given an algebraic expression {(a+2)$}$ and asked to find an equivalent expression when {a=\beta$}$. This means we need to substitute the value of {a$}$ with {\beta$}$ in the given expression and simplify it to find the equivalent expression.

Substitution Method

To solve this problem, we will use the substitution method. This method involves replacing the variable {a$}$ with its equivalent value {\beta$}$ in the given expression.

Step 1: Substitute {a$}$ with {\beta$}$

The given expression is {(a+2)$}$. We need to substitute {a$}$ with {\beta$}$ in this expression.

{(a+2) = (\beta+2)$}$

Step 2: Simplify the Expression

Now that we have substituted {a$}$ with {\beta$}$, we need to simplify the expression.

{(\beta+2)$}$

This expression is already simplified, but we can rewrite it in a different form to make it easier to compare with the given options.

{\beta+2$}$

Step 3: Compare with the Given Options

Now that we have simplified the expression, we can compare it with the given options to find the equivalent expression.

Option A: ${7(6)\$}

This option is not equivalent to the simplified expression {\beta+2$}$. The expression ${7(6)\$} is equal to ${42\$}, which is not the same as {\beta+2$}$.

Option B: ${9(6)\$}

This option is also not equivalent to the simplified expression {\beta+2$}$. The expression ${9(6)\$} is equal to ${54\$}, which is not the same as {\beta+2$}$.

Option C: ${7(\beta)+2\$}

This option is equivalent to the simplified expression {\beta+2$}$. The expression ${7(\beta)+2\$} is equal to ${7\beta+2\$}, which is the same as {\beta+2$}$.

Option D: ${7(6)+14\$}

This option is not equivalent to the simplified expression {\beta+2$}$. The expression ${7(6)+14\$} is equal to ${50\$}, which is not the same as {\beta+2$}$.

Conclusion

Based on the comparison with the given options, we can conclude that the equivalent expression to {(a+2)$}$ when {a=\beta$}$ is Option C: ${7(\beta)+2\$}.

Final Answer

The final answer is Option C: ${7(\beta)+2\$}.

Why is this the correct answer?

This is the correct answer because we have substituted {a$}$ with {\beta$}$ in the given expression and simplified it to find the equivalent expression. The simplified expression is {\beta+2$}$, which is the same as $7(\beta)+2\$}. Therefore, **Option C ${$7(\beta)+2$$** is the correct answer.

What is the significance of this problem?

This problem is significant because it demonstrates the importance of substitution and simplification in algebraic expressions. By substituting {a$}$ with {\beta$}$ and simplifying the expression, we can find the equivalent expression and solve the problem. This problem also highlights the need to carefully compare the given options with the simplified expression to find the correct answer.

What are the key concepts in this problem?

The key concepts in this problem are:

  • Substitution method
  • Simplification of algebraic expressions
  • Comparison of expressions
  • Equivalent expressions

What are the applications of this problem?

The applications of this problem are:

  • Algebraic expressions
  • Substitution method
  • Simplification of expressions
  • Equivalent expressions

What are the limitations of this problem?

The limitations of this problem are:

  • The problem assumes that the reader is familiar with algebraic expressions and substitution method.
  • The problem does not provide any real-world examples or applications of the concepts.
  • The problem is limited to a specific type of expression and does not cover other types of expressions.

What are the future directions of this problem?

The future directions of this problem are:

  • To provide more real-world examples and applications of the concepts.
  • To cover other types of expressions and provide more complex problems.
  • To develop more advanced techniques for simplifying and solving algebraic expressions.
    Frequently Asked Questions (FAQs) =====================================

Q: What is the substitution method in algebra?

A: The substitution method is a technique used in algebra to solve equations and expressions by substituting one variable with its equivalent value.

Q: How do I use the substitution method to solve an equation?

A: To use the substitution method, you need to:

  1. Identify the variable you want to substitute.
  2. Find its equivalent value.
  3. Substitute the variable with its equivalent value in the equation.
  4. Simplify the equation to find the solution.

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents a value that can change, while a constant is a value that remains the same.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to:

  1. Combine like terms.
  2. Remove any parentheses.
  3. Simplify any fractions or decimals.
  4. Write the expression in its simplest form.

Q: What is an equivalent expression?

A: An equivalent expression is an expression that has the same value as another expression, but is written in a different form.

Q: How do I find an equivalent expression?

A: To find an equivalent expression, you need to:

  1. Simplify the original expression.
  2. Write the simplified expression in a different form.
  3. Check that the new expression has the same value as the original expression.

Q: What are some common algebraic expressions?

A: Some common algebraic expressions include:

  • Linear expressions: {ax+b$}$
  • Quadratic expressions: {ax^2+bx+c$}$
  • Polynomial expressions: {a_nxn+a_{n-1}x{n-1}+...+a_1x+a_0$}$

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to:

  1. Factor the equation, if possible.
  2. Use the quadratic formula: {x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$}$
  3. Simplify the solutions to find the final answer.

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations of the form {ax^2+bx+c=0$}$. It is given by: {x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$}$

Q: How do I graph a linear equation?

A: To graph a linear equation, you need to:

  1. Find the x-intercept and y-intercept.
  2. Plot the intercepts on a coordinate plane.
  3. Draw a line through the intercepts to graph the equation.

Q: What is the difference between a function and a relation?

A: A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). A relation is a set of ordered pairs that satisfy a certain condition.

Q: How do I find the domain and range of a function?

A: To find the domain and range of a function, you need to:

  1. Identify the input values (domain).
  2. Identify the output values (range).
  3. Check that the function is defined for all input values.

Q: What is the significance of algebra in real-life situations?

A: Algebra is used in many real-life situations, including:

  • Science and engineering
  • Economics and finance
  • Computer programming and coding
  • Data analysis and statistics

Q: What are some common applications of algebra?

A: Some common applications of algebra include:

  • Solving systems of equations
  • Finding the maximum or minimum value of a function
  • Modeling real-world phenomena
  • Optimizing business decisions

Q: What are some common challenges in algebra?

A: Some common challenges in algebra include:

  • Solving complex equations
  • Finding the domain and range of a function
  • Graphing non-linear equations
  • Applying algebraic concepts to real-world problems

Q: How can I overcome these challenges?

A: To overcome these challenges, you need to:

  1. Practice regularly to build your skills and confidence.
  2. Seek help from teachers, tutors, or online resources.
  3. Break down complex problems into smaller, manageable parts.
  4. Use visual aids and diagrams to help you understand the concepts.