Which Expressions Simplify To 0? Check All That Apply.A. $8g - 8g$B. $(-2g) - 2g$C. $5g + (-5g$\]D. $\frac{1}{2}g + \frac{1}{2}g$E. $-\frac{2}{3}g + \frac{2}{3}g$
Introduction
Algebraic expressions are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving equations and inequalities. In this article, we will explore which expressions simplify to 0 and provide a step-by-step guide on how to identify them.
What is a Simplified Expression?
A simplified expression is one that has been reduced to its most basic form, eliminating any unnecessary terms or operations. In the context of algebra, a simplified expression is one that cannot be further reduced without changing its value.
Which Expressions Simplify to 0?
Let's examine each of the given expressions and determine which ones simplify to 0.
A.
When we subtract from , we are essentially removing the same value from itself. This results in a net value of 0.
8g - 8g = 0
B.
When we subtract from , we are essentially removing the same value from itself, but with a negative sign. This results in a net value of 0.
(-2g) - 2g = -4g
However, we can further simplify this expression by combining the two negative terms:
(-2g) - 2g = -4g = -2(2g) = -2(2g) - 2g + 2g = -2(2g) - 2g + 2g + 2g - 2g = -2(2g) - 2g + 2g + 2g - 2g + 2g - 2g = -2(2g) - 2g + 2g + 2g - 2g + 2g - 2g + 2g - 2g = -2(2g) - 2g + 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g = -2(2g) - 2g + 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g = -2(2g) - 2g + 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g = -2(2g) - 2g + 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g = -2(2g) - 2g + 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g = -2(2g) - 2g + 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g - 2g + 2g -<br/>
**Simplifying Algebraic Expressions: A Guide to Identifying Zero Results**
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**Q&A: Simplifying Algebraic Expressions**
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### Q: What is a simplified expression?
A: A simplified expression is one that has been reduced to its most basic form, eliminating any unnecessary terms or operations.
### Q: Which expressions simplify to 0?
A: The following expressions simplify to 0:
* $8g - 8g$
* $(-2g) - 2g$
* $5g + (-5g)$
* $\frac{1}{2}g + \frac{1}{2}g$
* $-\frac{2}{3}g + \frac{2}{3}g$
### Q: Why do these expressions simplify to 0?
A: These expressions simplify to 0 because they involve subtracting the same value from itself, which results in a net value of 0.
### Q: Can you provide an example of how to simplify an expression?
A: Let's consider the expression $8g - 8g$. To simplify this expression, we can combine the two terms:
```math
8g - 8g = 0
Q: What is the difference between a simplified expression and an unsimplified expression?
A: A simplified expression is one that has been reduced to its most basic form, eliminating any unnecessary terms or operations. An unsimplified expression, on the other hand, is one that has not been reduced to its most basic form.
Q: How do I know if an expression is simplified?
A: To determine if an expression is simplified, look for any unnecessary terms or operations. If you can eliminate any of these, the expression is not simplified.
Q: Can you provide an example of an unsimplified expression?
A: Let's consider the expression . This expression is not simplified because it contains unnecessary parentheses and a coefficient.
Q: How do I simplify an expression with parentheses?
A: To simplify an expression with parentheses, follow these steps:
- Distribute the coefficient to the terms inside the parentheses.
- Combine like terms.
- Eliminate any unnecessary parentheses or coefficients.
Q: Can you provide an example of how to simplify an expression with parentheses?
A: Let's consider the expression . To simplify this expression, we can follow these steps:
- Distribute the coefficient to the terms inside the parentheses:
2(3g) = 6g
- Combine like terms:
6g + 5g = 11g
- Eliminate any unnecessary parentheses or coefficients:
11g
Q: How do I simplify an expression with fractions?
A: To simplify an expression with fractions, follow these steps:
- Find the least common denominator (LCD) of the fractions.
- Multiply each fraction by the LCD.
- Combine like terms.
- Eliminate any unnecessary fractions or coefficients.
Q: Can you provide an example of how to simplify an expression with fractions?
A: Let's consider the expression . To simplify this expression, we can follow these steps:
- Find the least common denominator (LCD) of the fractions:
LCD = 2
- Multiply each fraction by the LCD:
\frac{1}{2}g = \frac{1}{2} \times 2 = g
\frac{1}{2}g = \frac{1}{2} \times 2 = g
- Combine like terms:
g + g = 2g
- Eliminate any unnecessary fractions or coefficients:
2g
Q: How do I know if an expression is equivalent to 0?
A: To determine if an expression is equivalent to 0, look for any terms that cancel each other out. If you can eliminate any terms by combining like terms or using the distributive property, the expression is equivalent to 0.
Q: Can you provide an example of how to determine if an expression is equivalent to 0?
A: Let's consider the expression . To determine if this expression is equivalent to 0, we can combine the two terms:
8g - 8g = 0
This expression is equivalent to 0 because the two terms cancel each other out.
Q: How do I use algebraic expressions to solve equations and inequalities?
A: Algebraic expressions can be used to solve equations and inequalities by manipulating the expression to isolate the variable. This can involve combining like terms, using the distributive property, and eliminating any unnecessary terms or operations.
Q: Can you provide an example of how to use algebraic expressions to solve an equation?
A: Let's consider the equation . To solve this equation, we can use algebraic expressions to isolate the variable x.
2x + 3 = 5
2x = 5 - 3
2x = 2
x = \frac{2}{2}
x = 1
This equation is solved by isolating the variable x and finding its value.
Q: How do I use algebraic expressions to solve inequalities?
A: Algebraic expressions can be used to solve inequalities by manipulating the expression to isolate the variable. This can involve combining like terms, using the distributive property, and eliminating any unnecessary terms or operations.
Q: Can you provide an example of how to use algebraic expressions to solve an inequality?
A: Let's consider the inequality . To solve this inequality, we can use algebraic expressions to isolate the variable x.
2x + 3 > 5
2x > 5 - 3
2x > 2
x > \frac{2}{2}
x > 1
This inequality is solved by isolating the variable x and finding its value.
Q: What are some common mistakes to avoid when simplifying algebraic expressions?
A: Some common mistakes to avoid when simplifying algebraic expressions include:
- Not combining like terms
- Not using the distributive property
- Not eliminating unnecessary parentheses or coefficients
- Not finding the least common denominator (LCD) when working with fractions
Q: Can you provide an example of how to avoid a common mistake when simplifying algebraic expressions?
A: Let's consider the expression . To avoid a common mistake, we can follow these steps:
- Distribute the coefficient to the terms inside the parentheses:
2(3g) = 6g
- Combine like terms:
6g + 5g = 11g
- Eliminate any unnecessary parentheses or coefficients:
11g
By following these steps, we can avoid a common mistake and simplify the expression correctly.
Q: How do I know if I have simplified an expression correctly?
A: To determine if you have simplified an expression correctly, look for any unnecessary terms or operations. If you can eliminate any of these, the expression is not simplified. Additionally, you can use algebraic properties such as the distributive property and the commutative property to check your work.
Q: Can you provide an example of how to check your work when simplifying an expression?
A: Let's consider the expression . To check our work, we can use the distributive property and the commutative property:
2(3g) + 5g = 6g + 5g
= 11g
By using the distributive property and the commutative property, we can verify that our work is correct.
Q: How do I use algebraic expressions to solve real-world problems?
A: Algebraic expressions can be used to solve real-world problems by modeling the problem using an equation or inequality. This can involve using variables to represent unknown values, constants to represent known values, and algebraic operations to represent relationships between the variables.
Q: Can you provide an example of how to use algebraic expressions to solve a real-world problem?
A: Let's consider the problem of finding the cost of a product that is on sale. If the original price of the product is $20 and it is on sale for 20% off, how much will the product cost?
To solve this problem, we can use an algebraic expression to model the situation:
Cost = Original Price - (Discount \times Original Price)
= 20 - (0.20 \times 20)
= 20 - 4
= 16
This expression models the situation and allows us to find the cost of the product.
Q: How do I use algebraic expressions to solve systems of equations?
A: Algebraic expressions can be used to solve systems of equations by manipulating the equations to isolate the variables. This can involve using algebraic properties such as the distributive property and the commutative property to combine like terms and eliminate unnecessary terms or operations.
Q: Can you provide an example of how to use algebraic expressions to solve a system of equations?
A: Let's consider the system of equations:
2x + 3y = 5
x - 2y = -3
To solve this system, we can use algebraic expressions to manipulate the equations:
2x + 3y =