In Exercises 27-34, Find The Difference.27. \[$(d-9)-(3d-1)\$\]28. \[$(6x+9)-(7x+1)\$\]29. \[$\left(y^2-4y+9\right)-\left(3y^2-6y-9\right)\$\]30. \[$\left(4m^2-m+2\right)-\left(-3m^2+10m+4\right)\$\]31.

Understanding the Concept of Difference

In mathematics, the difference between two expressions is a fundamental concept that is used to find the result of subtracting one expression from another. In this article, we will explore the concept of difference and learn how to find the difference between various algebraic expressions.

Exercise 27: Difference of Two Linear Expressions

The first exercise is to find the difference between the expressions $(d-9)$ and $(3d-1)$. To do this, we need to subtract the second expression from the first expression.

Step 1: Distribute the Negative Sign

When we subtract an expression, we need to distribute the negative sign to each term in the expression. In this case, we need to distribute the negative sign to the term $3d$ in the expression $(3d-1)$.

(d-9) - (3d-1) = d - 9 - 3d + 1

Step 2: Combine Like Terms

Now that we have distributed the negative sign, we can combine like terms. In this case, we can combine the terms $d$ and $-3d$.

d - 9 - 3d + 1 = -2d - 8

Therefore, the difference between the expressions $(d-9)$ and $(3d-1)$ is $-2d - 8$.

Exercise 28: Difference of Two Linear Expressions

The second exercise is to find the difference between the expressions $(6x+9)$ and $(7x+1)$. To do this, we need to subtract the second expression from the first expression.

Step 1: Distribute the Negative Sign

When we subtract an expression, we need to distribute the negative sign to each term in the expression. In this case, we need to distribute the negative sign to the term $7x$ in the expression $(7x+1)$.

(6x+9) - (7x+1) = 6x + 9 - 7x - 1

Step 2: Combine Like Terms

Now that we have distributed the negative sign, we can combine like terms. In this case, we can combine the terms $6x$ and $-7x$.

6x + 9 - 7x - 1 = -x + 8

Therefore, the difference between the expressions $(6x+9)$ and $(7x+1)$ is $-x + 8$.

Exercise 29: Difference of Two Quadratic Expressions

The third exercise is to find the difference between the expressions $\left(y^2-4y+9\right)$ and $\left(3y^2-6y-9\right)$. To do this, we need to subtract the second expression from the first expression.

Step 1: Distribute the Negative Sign

When we subtract an expression, we need to distribute the negative sign to each term in the expression. In this case, we need to distribute the negative sign to the terms $3y^2$, $-6y$, and $-9$ in the expression $\left(3y^2-6y-9\right)$.

\left(y^2-4y+9\right) - \left(3y^2-6y-9\right) = y^2 - 4y + 9 - 3y^2 + 6y + 9

Step 2: Combine Like Terms

Now that we have distributed the negative sign, we can combine like terms. In this case, we can combine the terms $y^2$ and $-3y^2$, and the terms $-4y$ and $6y$.

y^2 - 4y + 9 - 3y^2 + 6y + 9 = -2y^2 + 2y + 18

Therefore, the difference between the expressions $\left(y^2-4y+9\right)$ and $\left(3y^2-6y-9\right)$ is $-2y^2 + 2y + 18$.

Exercise 30: Difference of Two Quadratic Expressions

The fourth exercise is to find the difference between the expressions $\left(4m^2-m+2\right)$ and $\left(-3m^2+10m+4\right)$. To do this, we need to subtract the second expression from the first expression.

Step 1: Distribute the Negative Sign

When we subtract an expression, we need to distribute the negative sign to each term in the expression. In this case, we need to distribute the negative sign to the terms $-3m^2$, $10m$, and $4$ in the expression $\left(-3m^2+10m+4\right)$.

\left(4m^2-m+2\right) - \left(-3m^2+10m+4\right) = 4m^2 - m + 2 + 3m^2 - 10m - 4

Step 2: Combine Like Terms

Now that we have distributed the negative sign, we can combine like terms. In this case, we can combine the terms $4m^2$ and $3m^2$, and the terms $-m$ and $-10m$.

4m^2 - m + 2 + 3m^2 - 10m - 4 = 7m^2 - 11m - 2

Therefore, the difference between the expressions $\left(4m^2-m+2\right)$ and $\left(-3m^2+10m+4\right)$ is $7m^2 - 11m - 2$.

Conclusion

In this article, we have learned how to find the difference between various algebraic expressions. We have used the concept of distributing the negative sign and combining like terms to simplify the expressions. By following these steps, we can find the difference between any two algebraic expressions.

Key Takeaways

• To find the difference between two expressions, we need to subtract the second expression from the first expression.
• When we subtract an expression, we need to distribute the negative sign to each term in the expression.
• We can combine like terms to simplify the expression.

Final Thoughts

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about finding the difference between algebraic expressions.

Q: What is the difference between two algebraic expressions?

A: The difference between two algebraic expressions is the result of subtracting one expression from another. It is a fundamental concept in mathematics that is used to find the result of subtracting one expression from another.

Q: How do I find the difference between two algebraic expressions?

A: To find the difference between two algebraic expressions, you need to follow these steps:

1. Distribute the negative sign to each term in the second expression.
2. Combine like terms to simplify the expression.

Q: What is the difference between $(d-9)$ and $(3d-1)$?

A: To find the difference between $(d-9)$ and $(3d-1)$, we need to distribute the negative sign to each term in the second expression.

(d-9) - (3d-1) = d - 9 - 3d + 1

Then, we can combine like terms to simplify the expression.

d - 9 - 3d + 1 = -2d - 8

Therefore, the difference between $(d-9)$ and $(3d-1)$ is $-2d - 8$.

Q: What is the difference between $(6x+9)$ and $(7x+1)$?

A: To find the difference between $(6x+9)$ and $(7x+1)$, we need to distribute the negative sign to each term in the second expression.

(6x+9) - (7x+1) = 6x + 9 - 7x - 1

Then, we can combine like terms to simplify the expression.

6x + 9 - 7x - 1 = -x + 8

Therefore, the difference between $(6x+9)$ and $(7x+1)$ is $-x + 8$.

Q: What is the difference between $\left(y^2-4y+9\right)$ and $\left(3y^2-6y-9\right)$?

A: To find the difference between $\left(y^2-4y+9\right)$ and $\left(3y^2-6y-9\right)$, we need to distribute the negative sign to each term in the second expression.

\left(y^2-4y+9\right) - \left(3y^2-6y-9\right) = y^2 - 4y + 9 - 3y^2 + 6y + 9

Then, we can combine like terms to simplify the expression.

y^2 - 4y + 9 - 3y^2 + 6y + 9 = -2y^2 + 2y + 18

Therefore, the difference between $\left(y^2-4y+9\right)$ and $\left(3y^2-6y-9\right)$ is $-2y^2 + 2y + 18$.

Q: What is the difference between $\left(4m^2-m+2\right)$ and $\left(-3m^2+10m+4\right)$?

A: To find the difference between $\left(4m^2-m+2\right)$ and $\left(-3m^2+10m+4\right)$, we need to distribute the negative sign to each term in the second expression.

\left(4m^2-m+2\right) - \left(-3m^2+10m+4\right) = 4m^2 - m + 2 + 3m^2 - 10m - 4

Then, we can combine like terms to simplify the expression.

4m^2 - m + 2 + 3m^2 - 10m - 4 = 7m^2 - 11m - 2

Therefore, the difference between $\left(4m^2-m+2\right)$ and $\left(-3m^2+10m+4\right)$ is $7m^2 - 11m - 2$.

Conclusion

In this article, we have answered some of the most frequently asked questions about finding the difference between algebraic expressions. We have provided step-by-step solutions to each question, and we have explained the concept of distributing the negative sign and combining like terms.

Key Takeaways

• To find the difference between two algebraic expressions, you need to distribute the negative sign to each term in the second expression.
• You can combine like terms to simplify the expression.
• The difference between two algebraic expressions is a fundamental concept in mathematics that is used to find the result of subtracting one expression from another.

Final Thoughts

Finding the difference between algebraic expressions is an essential concept in mathematics. By understanding how to distribute the negative sign and combine like terms, you can simplify complex expressions and solve problems with ease. Whether you are a student or a professional, mastering this concept will help you to tackle a wide range of mathematical problems with confidence.