Simplify The Expression: $3x^2 - 12xy - 5x + 20y$

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. It involves combining like terms and rearranging the expression to make it more manageable and easier to work with. In this article, we will simplify the given expression: $3x^2 - 12xy - 5x + 20y$. We will use various techniques to combine like terms and rearrange the expression to its simplest form.

Understanding the Expression

The given expression is a quadratic expression in two variables, x and y. It consists of four terms: $3x^2$, $-12xy$, $-5x$, and $20y$. To simplify the expression, we need to identify the like terms and combine them.

Like Terms

Like terms are terms that have the same variable(s) raised to the same power. In this expression, the like terms are:

• $3x^2$ and $-5x$ (both have the variable x raised to the power of 2)
• $-12xy$ and $20y$ (both have the variable y raised to the power of 1)

Combining Like Terms

To combine like terms, we need to add or subtract the coefficients of the like terms. The coefficient of a term is the numerical value that multiplies the variable(s).

Combining $3x^2$ and $-5x$

To combine $3x^2$ and $-5x$, we need to add their coefficients. However, we cannot add these two terms because they have different variables raised to different powers. Therefore, we cannot combine them.

Combining $-12xy$ and $20y$

To combine $-12xy$ and $20y$, we need to add their coefficients. However, we cannot add these two terms because they have different variables raised to different powers. Therefore, we cannot combine them.

Rearranging the Expression

Since we cannot combine the like terms, we will rearrange the expression to group the like terms together.

Grouping Like Terms

We can group the like terms as follows:

• $3x^2$ (no like terms to combine)
• $-12xy$ (no like terms to combine)
• $-5x$ (no like terms to combine)
• $20y$ (no like terms to combine)

Final Simplified Expression

After rearranging the expression, we get:

$3x^2 - 12xy - 5x + 20y$

This is the final simplified expression.

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics. In this article, we simplified the given expression: $3x^2 - 12xy - 5x + 20y$. We identified the like terms and attempted to combine them, but were unable to do so because they had different variables raised to different powers. We then rearranged the expression to group the like terms together. The final simplified expression is: $3x^2 - 12xy - 5x + 20y$.

Additional Tips and Tricks

• When simplifying algebraic expressions, always look for like terms and combine them.
• Use the distributive property to expand expressions and combine like terms.
• Use the commutative and associative properties to rearrange expressions and combine like terms.
• Always check your work by plugging in values for the variables and simplifying the expression.

Frequently Asked Questions

• Q: What are like terms? A: Like terms are terms that have the same variable(s) raised to the same power.
• Q: How do I combine like terms? A: To combine like terms, add or subtract the coefficients of the like terms.
• Q: Can I combine terms with different variables raised to different powers? A: No, you cannot combine terms with different variables raised to different powers.

References

• [1] Algebra and Calculus, by Michael Artin
• [2] Mathematics for Dummies, by Mary Jane Sterling
• [3] Algebra and Trigonometry, by Michael Sullivan

Final Thoughts

Simplifying algebraic expressions is a crucial skill in mathematics. By understanding like terms and combining them, we can simplify expressions and make them more manageable. In this article, we simplified the given expression: $3x^2 - 12xy - 5x + 20y$. We identified the like terms and attempted to combine them, but were unable to do so because they had different variables raised to different powers. We then rearranged the expression to group the like terms together. The final simplified expression is: $3x^2 - 12xy - 5x + 20y$.

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Introduction

In our previous article, we simplified the given expression: $3x^2 - 12xy - 5x + 20y$. We identified the like terms and attempted to combine them, but were unable to do so because they had different variables raised to different powers. We then rearranged the expression to group the like terms together. The final simplified expression is: $3x^2 - 12xy - 5x + 20y$. In this article, we will answer some frequently asked questions related to simplifying algebraic expressions.

Q&A

Q: What are like terms?

A: Like terms are terms that have the same variable(s) raised to the same power. For example, $2x^2$ and $5x^2$ are like terms because they both have the variable x raised to the power of 2.

Q: How do I combine like terms?

A: To combine like terms, add or subtract the coefficients of the like terms. For example, if we have the expression $2x^2 + 5x^2$, we can combine the like terms by adding the coefficients: $2x^2 + 5x^2 = 7x^2$.

Q: Can I combine terms with different variables raised to different powers?

A: No, you cannot combine terms with different variables raised to different powers. For example, $2x^2$ and $3y^2$ are not like terms because they have different variables raised to different powers.

Q: What is the distributive property?

A: The distributive property is a property of algebra that states that a single term can be distributed to multiple terms. For example, if we have the expression $2(x + y)$, we can use the distributive property to expand it: $2(x + y) = 2x + 2y$.

Q: What is the commutative property?

A: The commutative property is a property of algebra that states that the order of the terms does not change the result. For example, if we have the expression $2x + 3y$, we can use the commutative property to rearrange it: $2x + 3y = 3y + 2x$.

Q: What is the associative property?

A: The associative property is a property of algebra that states that the order in which we perform operations does not change the result. For example, if we have the expression $(2x + 3y) + 4z$, we can use the associative property to rearrange it: $(2x + 3y) + 4z = 2x + (3y + 4z)$.

Q: How do I simplify an expression with multiple variables?

A: To simplify an expression with multiple variables, identify the like terms and combine them. Use the distributive property, commutative property, and associative property to rearrange the expression and combine like terms.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

• Not identifying like terms
• Not combining like terms correctly
• Not using the distributive property, commutative property, and associative property correctly
• Not checking the work by plugging in values for the variables

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics. By understanding like terms and combining them, we can simplify expressions and make them more manageable. In this article, we answered some frequently asked questions related to simplifying algebraic expressions. We hope that this article has been helpful in clarifying some common concepts and techniques in algebra.

Additional Tips and Tricks

• When simplifying algebraic expressions, always look for like terms and combine them.
• Use the distributive property to expand expressions and combine like terms.
• Use the commutative and associative properties to rearrange expressions and combine like terms.
• Always check your work by plugging in values for the variables.
• Use a calculator or computer program to check your work and ensure that you have simplified the expression correctly.

Frequently Asked Questions

• Q: What are some common mistakes to avoid when simplifying expressions? A: Some common mistakes to avoid when simplifying expressions include not identifying like terms, not combining like terms correctly, not using the distributive property, commutative property, and associative property correctly, and not checking the work by plugging in values for the variables.
• Q: How do I simplify an expression with multiple variables? A: To simplify an expression with multiple variables, identify the like terms and combine them. Use the distributive property, commutative property, and associative property to rearrange the expression and combine like terms.
• Q: What is the distributive property? A: The distributive property is a property of algebra that states that a single term can be distributed to multiple terms.
• Q: What is the commutative property? A: The commutative property is a property of algebra that states that the order of the terms does not change the result.
• Q: What is the associative property? A: The associative property is a property of algebra that states that the order in which we perform operations does not change the result.

References

• [1] Algebra and Calculus, by Michael Artin
• [2] Mathematics for Dummies, by Mary Jane Sterling
• [3] Algebra and Trigonometry, by Michael Sullivan

Final Thoughts

Simplifying algebraic expressions is a crucial skill in mathematics. By understanding like terms and combining them, we can simplify expressions and make them more manageable. In this article, we answered some frequently asked questions related to simplifying algebraic expressions. We hope that this article has been helpful in clarifying some common concepts and techniques in algebra.