Let's analyze each equation and determine which property of mathematics it exemplifies:
### a. \(-8 + 0 = -8\)
This equation shows that adding zero to any number does not change the value of that number. This is an example of the Additive Identity Property. Therefore:
[tex]\[ \text{a} = 1 \][/tex]
### b. \(5(x - 4) = 5x - 20\)
This equation distributes the multiplication of 5 to both \(x\) and \(-4\). This exemplifies the Distributive Property. Therefore:
[tex]\[ \text{b} = 5 \][/tex]
### c. \(5 \cdot \frac{1}{5} = 1\)
Here, multiplying a number by its reciprocal (multiplicative inverse) results in 1. This is an example of the Multiplicative Inverse Property. Therefore:
[tex]\[ \text{c} = 2 \][/tex]
### d. \((3 + 5) + 9 = 3 + (5 + 9)\)
This equation shows that the way in which numbers are grouped when adding does not affect the sum. This is known as the Associative Property of Addition. Therefore:
[tex]\[ \text{d} = 6 \][/tex]
### e. \(7 + (-7) = 0\)
This equation shows that adding a number to its additive inverse results in zero. This illustrates the Additive Inverse Property. Therefore:
[tex]\[ \text{e} = 3 \][/tex]
### f. \(5 \times 6 = 6 \times 5\)
This equation demonstrates that the order in which numbers are multiplied does not affect the product. This is known as the Commutative Property of Multiplication. Therefore:
[tex]\[ \text{f} = 4 \][/tex]
### Summary
By matching each equation with the corresponding property, we have:
- (a) \(-8 + 0 = -8\) corresponds to the Additive Identity Property, which is property 1.
- (b) \(5(x - 4) = 5x - 20\) corresponds to the Distributive Property, which is property 5.
- (c) \(5 \cdot \frac{1}{5} = 1\) corresponds to the Multiplicative Inverse Property, which is property 2.
- (d) \((3 + 5) + 9 = 3 + (5 + 9)\) corresponds to the Associative Property of Addition, which is property 6.
- (e) \(7 + (-7) = 0\) corresponds to the Additive Inverse Property, which is property 3.
- (f) \(5 \times 6 = 6 \times 5\) corresponds to the Commutative Property of Multiplication, which is property 4.
Thus, the matches are:
[tex]\[ \{\text{a}: 1, \text{b}: 5, \text{c}: 2, \text{d}: 6, \text{e}: 3, \text{f}: 4\} \][/tex]
