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To determine which statements correctly illustrate the addition property of equality, let's examine each statement in detail.
### Understanding the Addition Property of Equality
The addition property of equality states that if you add the same quantity to both sides of an equation, the equality remains true. Mathematically, if [tex]\( a = b \)[/tex], then [tex]\( a + c = b + c \)[/tex] for any number [tex]\( c \)[/tex]. Similarly, if you subtract the same quantity from both sides, the equality remains true.
### Analyzing Each Statement
1. Statement: If [tex]\( a = b \)[/tex], then [tex]\( a + c = b + c \)[/tex].
- Analysis: This directly follows the addition property of equality. Adding [tex]\( c \)[/tex] to both sides of the equation does not change the equality.
- Conclusion: This statement is correct.
2. Statement: If [tex]\( x = y \)[/tex], then [tex]\( x + 2 = y - 2 \)[/tex].
- Analysis: This does not follow the addition property of equality because we are adding different quantities to each side of the equation. Specifically, adding 2 to the left side and subtracting 2 from the right side changes the equality.
- Conclusion: This statement is incorrect.
3. Statement: If [tex]\( w + 2 = 7 \)[/tex], then [tex]\( w + 2 - 2 = 7 - 2 \)[/tex].
- Analysis: Here, we are subtracting the same quantity (2) from both sides of the equation, which follows the addition property of equality (subtraction is essentially adding a negative number).
- Conclusion: This statement is correct.
4. Statement: If [tex]\( z - \frac{2}{5} = 9 \)[/tex], then [tex]\( z - \frac{2}{5} + \frac{2}{5} = 9 - \frac{2}{5} \)[/tex].
- Analysis: In this case, we are adding [tex]\(\frac{2}{5}\)[/tex] to both sides of the equation, which maintains the equality. This follows the addition property of equality.
- Conclusion: This statement is correct.
### Summary
Based on the analysis, the statements that correctly illustrate the addition property of equality are:
- If [tex]\( a = b \)[/tex], then [tex]\( a + c = b + c \)[/tex].
- If [tex]\( w + 2 = 7 \)[/tex], then [tex]\( w + 2 - 2 = 7 - 2 \)[/tex].
- If [tex]\( z - \frac{2}{5} = 9 \)[/tex], then [tex]\( z - \frac{2}{5} + \frac{2}{5} = 9 - \frac{2}{5} \)[/tex].
Thus, the correct statements are:
1. If [tex]\( a = b \)[/tex], then [tex]\( a + c = b + c \)[/tex].
3. If [tex]\( w + 2 = 7 \)[/tex], then [tex]\( w + 2 - 2 = 7 - 2 \)[/tex].
4. If [tex]\( z - \frac{2}{5} = 9 \)[/tex], then [tex]\( z - \frac{2}{5} + \frac{2}{5} = 9 - \frac{2}{5} \)[/tex].
### Understanding the Addition Property of Equality
The addition property of equality states that if you add the same quantity to both sides of an equation, the equality remains true. Mathematically, if [tex]\( a = b \)[/tex], then [tex]\( a + c = b + c \)[/tex] for any number [tex]\( c \)[/tex]. Similarly, if you subtract the same quantity from both sides, the equality remains true.
### Analyzing Each Statement
1. Statement: If [tex]\( a = b \)[/tex], then [tex]\( a + c = b + c \)[/tex].
- Analysis: This directly follows the addition property of equality. Adding [tex]\( c \)[/tex] to both sides of the equation does not change the equality.
- Conclusion: This statement is correct.
2. Statement: If [tex]\( x = y \)[/tex], then [tex]\( x + 2 = y - 2 \)[/tex].
- Analysis: This does not follow the addition property of equality because we are adding different quantities to each side of the equation. Specifically, adding 2 to the left side and subtracting 2 from the right side changes the equality.
- Conclusion: This statement is incorrect.
3. Statement: If [tex]\( w + 2 = 7 \)[/tex], then [tex]\( w + 2 - 2 = 7 - 2 \)[/tex].
- Analysis: Here, we are subtracting the same quantity (2) from both sides of the equation, which follows the addition property of equality (subtraction is essentially adding a negative number).
- Conclusion: This statement is correct.
4. Statement: If [tex]\( z - \frac{2}{5} = 9 \)[/tex], then [tex]\( z - \frac{2}{5} + \frac{2}{5} = 9 - \frac{2}{5} \)[/tex].
- Analysis: In this case, we are adding [tex]\(\frac{2}{5}\)[/tex] to both sides of the equation, which maintains the equality. This follows the addition property of equality.
- Conclusion: This statement is correct.
### Summary
Based on the analysis, the statements that correctly illustrate the addition property of equality are:
- If [tex]\( a = b \)[/tex], then [tex]\( a + c = b + c \)[/tex].
- If [tex]\( w + 2 = 7 \)[/tex], then [tex]\( w + 2 - 2 = 7 - 2 \)[/tex].
- If [tex]\( z - \frac{2}{5} = 9 \)[/tex], then [tex]\( z - \frac{2}{5} + \frac{2}{5} = 9 - \frac{2}{5} \)[/tex].
Thus, the correct statements are:
1. If [tex]\( a = b \)[/tex], then [tex]\( a + c = b + c \)[/tex].
3. If [tex]\( w + 2 = 7 \)[/tex], then [tex]\( w + 2 - 2 = 7 - 2 \)[/tex].
4. If [tex]\( z - \frac{2}{5} = 9 \)[/tex], then [tex]\( z - \frac{2}{5} + \frac{2}{5} = 9 - \frac{2}{5} \)[/tex].
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