Let’s verify the given equations step by step.
### 1) Verify [tex]\(a + b = b + a\)[/tex]:
Given:
[tex]\[ a = \frac{-5}{6} \][/tex]
[tex]\[ b = \frac{3}{4} \][/tex]
First, we find [tex]\(a + b\)[/tex]:
[tex]\[ a + b = \frac{-5}{6} + \frac{3}{4} \][/tex]
To add these fractions, we need a common denominator. The least common multiple of 6 and 4 is 12. Thus:
[tex]\[ \frac{-5}{6} = \frac{-5 \cdot 2}{6 \cdot 2} = \frac{-10}{12} \][/tex]
[tex]\[ \frac{3}{4} = \frac{3 \cdot 3}{4 \cdot 3} = \frac{9}{12} \][/tex]
Now add the fractions:
[tex]\[ a + b = \frac{-10}{12} + \frac{9}{12} = \frac{-10 + 9}{12} = \frac{-1}{12} \][/tex]
Next, we find [tex]\(b + a\)[/tex]:
[tex]\[ b + a = \frac{3}{4} + \frac{-5}{6} \][/tex]
We already converted these fractions:
[tex]\[ \frac{3}{4} = \frac{9}{12} \][/tex]
[tex]\[ \frac{-5}{6} = \frac{-10}{12} \][/tex]
Now add them:
[tex]\[ b + a = \frac{9}{12} + \frac{-10}{12} = \frac{9 - 10}{12} = \frac{-1}{12} \][/tex]
Since:
[tex]\[ a + b = \frac{-1}{12} \][/tex]
[tex]\[ b + a = \frac{-1}{12} \][/tex]
We conclude that:
[tex]\[ a + b = b + a \][/tex]
### 2) Verify [tex]\(a \times b = b \times a\)[/tex]:
Given:
[tex]\[ a = \frac{-5}{6} \][/tex]
[tex]\[ b = \frac{3}{4} \][/tex]
First, we find [tex]\(a \times b\)[/tex]:
[tex]\[ a \times b = \frac{-5}{6} \times \frac{3}{4} \][/tex]
Multiply the numerators and the denominators:
[tex]\[ a \times b = \frac{-5 \cdot 3}{6 \cdot 4} = \frac{-15}{24} \][/tex]
Next, we find [tex]\(b \times a\)[/tex]:
[tex]\[ b \times a = \frac{3}{4} \times \frac{-5}{6} \][/tex]
Similarly, multiply the numerators and the denominators:
[tex]\[ b \times a = \frac{3 \cdot -5}{4 \cdot 6} = \frac{-15}{24} \][/tex]
Since:
[tex]\[ a \times b = \frac{-15}{24} \][/tex]
[tex]\[ b \times a = \frac{-15}{24} \][/tex]
We conclude that:
[tex]\[ a \times b = b \times a \][/tex]
So, both properties have been verified.
