To determine which statements correctly illustrate the addition property of equality, we'll examine each given statement.
1. If [tex]\(a = b\)[/tex], then [tex]\(a + c = b + c\)[/tex]:
- The addition property of equality states that if two expressions are equal, then adding the same value to both expressions maintains the equality. In this case, adding [tex]\(c\)[/tex] to both sides of the equation [tex]\(a = b\)[/tex] results in [tex]\(a + c = b + c\)[/tex]. This is a valid application of the addition property of equality.
2. If [tex]\(x = y\)[/tex], then [tex]\(x + 2 = y - 2\)[/tex]:
- Here, we have [tex]\(x = y\)[/tex]. To maintain equality, we must add or subtract the same value from both sides of the equation. However, adding 2 to one side and subtracting 2 from the other side, results in [tex]\(x + 2\)[/tex] and [tex]\(y - 2\)[/tex], which are not guaranteed to be equal if [tex]\(x = y\)[/tex]. This does not correctly illustrate the addition property of equality.
3. If [tex]\(w + 2 = 7\)[/tex], then [tex]\(w + 2 - 2 = 7 - 2\)[/tex]:
- Starting from the equation [tex]\(w + 2 = 7\)[/tex], subtracting the same value [tex]\(2\)[/tex] from both sides results in [tex]\(w + 2 - 2 = 7 - 2\)[/tex]. This simplifies to [tex]\(w = 5\)[/tex]. This is a correct application of the addition property of equality, demonstrating how equality is maintained when subtracting the same value from both sides.
4. If [tex]\(z - \frac{2}{5} = 9\)[/tex], then [tex]\(z - \frac{2}{5} + \frac{2}{5} = 9 - \frac{2}{5}\)[/tex]:
- From the equation [tex]\(z - \frac{2}{5} = 9\)[/tex], adding the same value [tex]\(\frac{2}{5}\)[/tex] to both sides results in [tex]\(z - \frac{2}{5} + \frac{2}{5} = 9 + \frac{2}{5}\)[/tex]. This simplifies to [tex]\(z = 9 + \frac{2}{5}\)[/tex]. This application correctly uses the addition property of equality by adding [tex]\(\frac{2}{5}\)[/tex] to both sides of the equation.
Therefore, the correct statements that illustrate the addition property of equality are:
- Statement 1: If [tex]\(a = b\)[/tex], then [tex]\(a + c = b + c\)[/tex].
- Statement 3: If [tex]\(w + 2 = 7\)[/tex], then [tex]\(w + 2 - 2 = 7 - 2\)[/tex].
- Statement 4: 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:
[tex]\[ \boxed{1, 3, 4} \][/tex]
