Find: $\left(4 X^2 Y^3 + 2 X Y^2 - 2 Y\right) - \left(-7 X^2 Y^3 + 6 X Y^2 - 2 Y\right$\]Place The Correct Coefficients In The Difference:$\square \, X^2 Y^3 + \square \, X Y^2 + \square \, Y$

Feb 26, 2025 by ADMIN 193 views

**Simplifying Algebraic Expressions: A Step-by-Step Guide**

===========================================================

Understanding the Problem

When simplifying algebraic expressions, it's essential to combine like terms and eliminate any unnecessary components. In this article, we'll focus on finding the difference between two given expressions and placing the correct coefficients in the resulting expression.

The Given Expressions

The two given expressions are:

$\left(4 x^2 y^3 + 2 x y^2 - 2 y\right)$

and

$\left(-7 x^2 y^3 + 6 x y^2 - 2 y\right)$

Combining Like Terms

To simplify the given expressions, we need to combine like terms. Like terms are terms that have the same variable(s) raised to the same power. In this case, we have three types of like terms:

Terms with $x^2 y^3$
Terms with $x y^2$
Terms with $y$

Combining Terms with $x^2 y^3$

The first expression has a term with $x^2 y^3$ and the second expression has a term with $x^2 y^3$ . To combine these terms, we add their coefficients:

$4 x^2 y^3 + (-7) x^2 y^3$

The coefficient of the first term is 4 and the coefficient of the second term is -7. When we add these coefficients, we get:

$-3 x^2 y^3$

Combining Terms with $x y^2$

The first expression has a term with $x y^2$ and the second expression has a term with $x y^2$ . To combine these terms, we add their coefficients:

$2 x y^2 + 6 x y^2$

The coefficient of the first term is 2 and the coefficient of the second term is 6. When we add these coefficients, we get:

$8 x y^2$

Combining Terms with $y$

The first expression has a term with $y$ and the second expression has a term with $y$ . To combine these terms, we add their coefficients:

$-2 y + (-2) y$

The coefficient of the first term is -2 and the coefficient of the second term is -2. When we add these coefficients, we get:

$-4 y$

The Simplified Expression

Now that we have combined all the like terms, we can write the simplified expression:

$-3 x^2 y^3 + 8 x y^2 - 4 y$

Conclusion

In this article, we simplified two given algebraic expressions by combining like terms. We identified the like terms, combined their coefficients, and eliminated any unnecessary components. The resulting expression is a simplified version of the original expressions.

Tips and Tricks

When simplifying algebraic expressions, it's essential to:

Identify like terms
Combine their coefficients
Eliminate any unnecessary components

By following these steps, you can simplify complex algebraic expressions and make them easier to work with.

Real-World Applications

Simplifying algebraic expressions has many real-world applications, including:

Physics: Simplifying expressions is essential in physics, where complex equations are used to describe the behavior of physical systems.
Engineering: Simplifying expressions is also crucial in engineering, where complex equations are used to design and analyze systems.
Computer Science: Simplifying expressions is also used in computer science, where complex algorithms are used to solve problems.

Final Thoughts

Simplifying algebraic expressions is an essential skill in mathematics and has many real-world applications. By following the steps outlined in this article, you can simplify complex expressions and make them easier to work with.

===========================================================

Q: What is the difference between combining like terms and simplifying an expression?

A: Combining like terms is the process of adding or subtracting terms that have the same variable(s) raised to the same power. Simplifying an expression, on the other hand, is the process of combining like terms and eliminating any unnecessary components to make the expression easier to work with.

Q: How do I identify like terms in an expression?

A: To identify like terms, look for terms that have the same variable(s) raised to the same power. For example, in the expression $2x^2 + 3x^2$ , the terms $2x^2$ and $3x^2$ are like terms because they both have the variable $x$ raised to the power of 2.

Q: Can I combine unlike terms?

A: No, unlike terms cannot be combined. Unlike terms are terms that have different variables or variables raised to different powers. For example, in the expression $2x + 3y$ , the terms $2x$ and $3y$ are unlike terms and cannot be combined.

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

A: To simplify an expression with multiple variables, identify the like terms and combine their coefficients. For example, in the expression $2x^2y + 3x^2y + 4xy$ , the terms $2x^2y$ and $3x^2y$ are like terms and can be combined to get $5x^2y$ . The term $4xy$ is an unlike term and cannot be combined with the other terms.

Q: Can I simplify an expression with negative coefficients?

A: Yes, you can simplify an expression with negative coefficients. When combining like terms with negative coefficients, remember to add the coefficients as you would with positive coefficients. For example, in the expression $-2x^2 + 3x^2$ , the coefficients are -2 and 3. When you add these coefficients, you get $1x^2$ .

Q: How do I know when an expression is simplified?

A: An expression is simplified when all like terms have been combined and there are no unnecessary components. For example, in the expression $2x^2 + 3x^2 + 4xy$ , the expression is not simplified because the terms $2x^2$ and $3x^2$ are like terms and can be combined to get $5x^2$ . The term $4xy$ is an unlike term and cannot be combined with the other terms.

Q: Can I simplify an expression with fractions?

A: Yes, you can simplify an expression with fractions. When combining like terms with fractions, remember to add the numerators and keep the common denominator. For example, in the expression $\frac{2}{3}x^2 + \frac{3}{3}x^2$ , the fractions have a common denominator of 3. When you add the numerators, you get $\frac{5}{3}x^2$ .

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, remember to follow the rules of exponentiation. For example, in the expression $x^2 \cdot x^3$ , the exponents can be combined to get $x^{2+3} = x^5$ .

Q: Can I simplify an expression with radicals?

A: Yes, you can simplify an expression with radicals. When combining like terms with radicals, remember to multiply the coefficients and add the indices. For example, in the expression $\sqrt{2} + \sqrt{3}$ , the radicals cannot be combined because they have different indices.

Q: How do I know if an expression is in its simplest form?

A: An expression is in its simplest form when all like terms have been combined and there are no unnecessary components. For example, in the expression $2x^2 + 3x^2 + 4xy$ , the expression is not in its simplest form because the terms $2x^2$ and $3x^2$ are like terms and can be combined to get $5x^2$ . The term $4xy$ is an unlike term and cannot be combined with the other terms.

Q: Can I simplify an expression with absolute values?

A: Yes, you can simplify an expression with absolute values. When combining like terms with absolute values, remember to remove the absolute value signs and simplify the expression. For example, in the expression $|2x| + |3x|$ , the absolute value signs can be removed to get $2x + 3x = 5x$ .

Q: How do I simplify an expression with inequalities?

A: To simplify an expression with inequalities, remember to follow the rules of inequality. For example, in the expression $2x + 3 > 5$ , the inequality can be simplified by subtracting 3 from both sides to get $2x > 2$ .