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Sagot :
Let's match each equation to its corresponding property step-by-step:
1. [tex]\((x + y) + z = x + (y + z)\)[/tex] is the Associative Property of Addition. This property states that how you group the numbers in addition does not change the result. Hence, it matches with the option [tex]\(d\)[/tex].
[tex]\[ \text{1: } d \][/tex]
2. [tex]\(0 \div 5 = 0 ; 5 \div 0 = \text{undefined}\)[/tex] refers to the Division Property of Zero. Dividing zero by any number yields zero, but dividing any number by zero is undefined. This matches with the option [tex]\(l\)[/tex].
[tex]\[ \text{2: } l \][/tex]
3. [tex]\(x \cdot 0 = 0\)[/tex] is the Multiplication Property of Zero. Any number multiplied by zero results in zero. This matches with the option [tex]\(g\)[/tex].
[tex]\[ \text{3: } g \][/tex]
4. [tex]\(x + (-x) = 0\)[/tex] is the Addition Property of Opposites. Any number added to its opposite (negative) equals zero. This matches with the option [tex]\(b\)[/tex].
[tex]\[ \text{4: } b \][/tex]
5. [tex]\((x \cdot y) \cdot z = x \cdot (y \cdot z)\)[/tex] is the Associative Property of Multiplication. This property states that how you group the numbers in multiplication does not change the result. Hence, it matches with the option [tex]\(j\)[/tex].
[tex]\[ \text{5: } j \][/tex]
6. [tex]\(x \cdot y = y \cdot x\)[/tex] is the Commutative Property of Multiplication. It states that changing the order of the numbers in multiplication does not change the result. This matches with the option [tex]\(i\)[/tex].
[tex]\[ \text{6: } i \][/tex]
7. [tex]\(x \div 1 = x\)[/tex] is the Division Property of One. Dividing any number by one yields the number itself. This matches with the option [tex]\(m\)[/tex].
[tex]\[ \text{7: } m \][/tex]
8. [tex]\(x + 0 = x\)[/tex] is the Addition Property of Zero. Adding zero to any number yields the number itself. This matches with the option [tex]\(a\)[/tex].
[tex]\[ \text{8: } a \][/tex]
9. [tex]\(x - 0 = x\)[/tex] is the Subtraction Property of Zero. Subtracting zero from any number yields the number itself. This matches with the option [tex]\(e\)[/tex].
[tex]\[ \text{9: } e \][/tex]
10. [tex]\(x + y = y + x\)[/tex] is the Commutative Property of Addition. It states that changing the order of the numbers in addition does not change the result. This matches with the option [tex]\(c\)[/tex].
[tex]\[ \text{10: } c \][/tex]
11. [tex]\(x(y + z) = xy + xz\)[/tex] is the Distributive Property of Multiplication. This property explains how multiplication distributes over addition. Hence, it matches with the option [tex]\(k\)[/tex].
[tex]\[ \text{11: } k \][/tex]
12. [tex]\(x \cdot 1 = x\)[/tex] is the Multiplication Property of One. Multiplying any number by one yields the number itself. This matches with the option [tex]\(h\)[/tex].
[tex]\[ \text{12: } h \][/tex]
13. [tex]\(x - y = x + (-y)\)[/tex] is the Subtraction Property of Regrouping. This property explains how subtraction can be represented as adding the opposite. Hence, it matches with the option [tex]\(f\)[/tex].
[tex]\[ \text{13: } f \][/tex]
So the final matching is:
[tex]\[ \{1: 'd', 2: 'l', 3: 'g', 4: 'b', 5: 'j', 6: 'i', 7: 'm', 8: 'a', 9: 'e', 10: 'c', 11: 'k', 12: 'h', 13: 'f'\} \][/tex]
1. [tex]\((x + y) + z = x + (y + z)\)[/tex] is the Associative Property of Addition. This property states that how you group the numbers in addition does not change the result. Hence, it matches with the option [tex]\(d\)[/tex].
[tex]\[ \text{1: } d \][/tex]
2. [tex]\(0 \div 5 = 0 ; 5 \div 0 = \text{undefined}\)[/tex] refers to the Division Property of Zero. Dividing zero by any number yields zero, but dividing any number by zero is undefined. This matches with the option [tex]\(l\)[/tex].
[tex]\[ \text{2: } l \][/tex]
3. [tex]\(x \cdot 0 = 0\)[/tex] is the Multiplication Property of Zero. Any number multiplied by zero results in zero. This matches with the option [tex]\(g\)[/tex].
[tex]\[ \text{3: } g \][/tex]
4. [tex]\(x + (-x) = 0\)[/tex] is the Addition Property of Opposites. Any number added to its opposite (negative) equals zero. This matches with the option [tex]\(b\)[/tex].
[tex]\[ \text{4: } b \][/tex]
5. [tex]\((x \cdot y) \cdot z = x \cdot (y \cdot z)\)[/tex] is the Associative Property of Multiplication. This property states that how you group the numbers in multiplication does not change the result. Hence, it matches with the option [tex]\(j\)[/tex].
[tex]\[ \text{5: } j \][/tex]
6. [tex]\(x \cdot y = y \cdot x\)[/tex] is the Commutative Property of Multiplication. It states that changing the order of the numbers in multiplication does not change the result. This matches with the option [tex]\(i\)[/tex].
[tex]\[ \text{6: } i \][/tex]
7. [tex]\(x \div 1 = x\)[/tex] is the Division Property of One. Dividing any number by one yields the number itself. This matches with the option [tex]\(m\)[/tex].
[tex]\[ \text{7: } m \][/tex]
8. [tex]\(x + 0 = x\)[/tex] is the Addition Property of Zero. Adding zero to any number yields the number itself. This matches with the option [tex]\(a\)[/tex].
[tex]\[ \text{8: } a \][/tex]
9. [tex]\(x - 0 = x\)[/tex] is the Subtraction Property of Zero. Subtracting zero from any number yields the number itself. This matches with the option [tex]\(e\)[/tex].
[tex]\[ \text{9: } e \][/tex]
10. [tex]\(x + y = y + x\)[/tex] is the Commutative Property of Addition. It states that changing the order of the numbers in addition does not change the result. This matches with the option [tex]\(c\)[/tex].
[tex]\[ \text{10: } c \][/tex]
11. [tex]\(x(y + z) = xy + xz\)[/tex] is the Distributive Property of Multiplication. This property explains how multiplication distributes over addition. Hence, it matches with the option [tex]\(k\)[/tex].
[tex]\[ \text{11: } k \][/tex]
12. [tex]\(x \cdot 1 = x\)[/tex] is the Multiplication Property of One. Multiplying any number by one yields the number itself. This matches with the option [tex]\(h\)[/tex].
[tex]\[ \text{12: } h \][/tex]
13. [tex]\(x - y = x + (-y)\)[/tex] is the Subtraction Property of Regrouping. This property explains how subtraction can be represented as adding the opposite. Hence, it matches with the option [tex]\(f\)[/tex].
[tex]\[ \text{13: } f \][/tex]
So the final matching is:
[tex]\[ \{1: 'd', 2: 'l', 3: 'g', 4: 'b', 5: 'j', 6: 'i', 7: 'm', 8: 'a', 9: 'e', 10: 'c', 11: 'k', 12: 'h', 13: 'f'\} \][/tex]
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