Answer :
To verify the equation [tex]\( a \times (b - c) = (a \times b) - (a \times c) \)[/tex] given [tex]\( a = \frac{-6}{7} \)[/tex], [tex]\( b = \frac{1}{3} \)[/tex], and [tex]\( c = \frac{-4}{15} \)[/tex], let's go through the mathematical steps in detail.
### Step 1: Calculate [tex]\( b - c \)[/tex]
First, we need to find the difference [tex]\( b - c \)[/tex].
[tex]\[ b - c = \frac{1}{3} - \frac{-4}{15} \][/tex]
To subtract these two fractions, we need a common denominator. The least common multiple of 3 and 15 is 15. Convert both fractions to have this common denominator:
[tex]\[ \frac{1}{3} = \frac{5}{15} \][/tex]
[tex]\[ \frac{-4}{15} = \frac{-4}{15} \][/tex]
Now we can subtract the fractions:
[tex]\[ b - c = \frac{5}{15} - \frac{-4}{15} = \frac{5}{15} + \frac{4}{15} = \frac{9}{15} = 0.6 \][/tex]
### Step 2: Calculate [tex]\( a \times (b - c) \)[/tex]
Next, we need to multiply [tex]\( a \)[/tex] by the result from the previous step:
[tex]\[ a \times (b - c) = \frac{-6}{7} \times 0.6 \][/tex]
We convert 0.6 to a fraction:
[tex]\[ 0.6 = \frac{6}{10} = \frac{3}{5} \][/tex]
So,
[tex]\[ a \times (b - c) = \frac{-6}{7} \times \frac{3}{5} = \frac{-18}{35} \approx -0.5142857142857142 \][/tex]
### Step 3: Calculate [tex]\( a \times b \)[/tex]
Now we calculate [tex]\( a \times b \)[/tex]:
[tex]\[ a \times b = \frac{-6}{7} \times \frac{1}{3} \][/tex]
[tex]\[ a \times b = \frac{-6}{21} = \frac{-2}{7} \approx -0.2857142857142857 \][/tex]
### Step 4: Calculate [tex]\( a \times c \)[/tex]
Next, we calculate [tex]\( a \times c \)[/tex]:
[tex]\[ a \times c = \frac{-6}{7} \times \frac{-4}{15} \][/tex]
[tex]\[ a \times c = \frac{24}{105} = \frac{8}{35} \approx 0.22857142857142856 \][/tex]
### Step 5: Calculate [tex]\( (a \times b) - (a \times c) \)[/tex]
Now we need to find the difference between [tex]\( a \times b \)[/tex] and [tex]\( a \times c \)[/tex]:
[tex]\[ (a \times b) - (a \times c) = \frac{-2}{7} - \frac{8}{35} \][/tex]
First, convert [tex]\(\frac{-2}{7}\)[/tex] to have a common denominator with [tex]\(\frac{8}{35}\)[/tex]:
[tex]\[ \frac{-2}{7} = \frac{-10}{35} \][/tex]
Now subtract the fractions:
[tex]\[ (a \times b) - (a \times c) = \frac{-10}{35} - \frac{8}{35} = \frac{-18}{35} \approx -0.5142857142857142 \][/tex]
### Step 6: Verification
At this point, we compare the results from Step 2 and Step 5:
[tex]\[ a \times (b - c) \approx -0.5142857142857142 \][/tex]
[tex]\[ (a \times b) - (a \times c) \approx -0.5142857142857142 \][/tex]
Since both results are approximately equal, we have verified that:
[tex]\[ a \times (b - c) = (a \times b) - (a \times c) \][/tex]
Thus, the equation holds true.
### Step 1: Calculate [tex]\( b - c \)[/tex]
First, we need to find the difference [tex]\( b - c \)[/tex].
[tex]\[ b - c = \frac{1}{3} - \frac{-4}{15} \][/tex]
To subtract these two fractions, we need a common denominator. The least common multiple of 3 and 15 is 15. Convert both fractions to have this common denominator:
[tex]\[ \frac{1}{3} = \frac{5}{15} \][/tex]
[tex]\[ \frac{-4}{15} = \frac{-4}{15} \][/tex]
Now we can subtract the fractions:
[tex]\[ b - c = \frac{5}{15} - \frac{-4}{15} = \frac{5}{15} + \frac{4}{15} = \frac{9}{15} = 0.6 \][/tex]
### Step 2: Calculate [tex]\( a \times (b - c) \)[/tex]
Next, we need to multiply [tex]\( a \)[/tex] by the result from the previous step:
[tex]\[ a \times (b - c) = \frac{-6}{7} \times 0.6 \][/tex]
We convert 0.6 to a fraction:
[tex]\[ 0.6 = \frac{6}{10} = \frac{3}{5} \][/tex]
So,
[tex]\[ a \times (b - c) = \frac{-6}{7} \times \frac{3}{5} = \frac{-18}{35} \approx -0.5142857142857142 \][/tex]
### Step 3: Calculate [tex]\( a \times b \)[/tex]
Now we calculate [tex]\( a \times b \)[/tex]:
[tex]\[ a \times b = \frac{-6}{7} \times \frac{1}{3} \][/tex]
[tex]\[ a \times b = \frac{-6}{21} = \frac{-2}{7} \approx -0.2857142857142857 \][/tex]
### Step 4: Calculate [tex]\( a \times c \)[/tex]
Next, we calculate [tex]\( a \times c \)[/tex]:
[tex]\[ a \times c = \frac{-6}{7} \times \frac{-4}{15} \][/tex]
[tex]\[ a \times c = \frac{24}{105} = \frac{8}{35} \approx 0.22857142857142856 \][/tex]
### Step 5: Calculate [tex]\( (a \times b) - (a \times c) \)[/tex]
Now we need to find the difference between [tex]\( a \times b \)[/tex] and [tex]\( a \times c \)[/tex]:
[tex]\[ (a \times b) - (a \times c) = \frac{-2}{7} - \frac{8}{35} \][/tex]
First, convert [tex]\(\frac{-2}{7}\)[/tex] to have a common denominator with [tex]\(\frac{8}{35}\)[/tex]:
[tex]\[ \frac{-2}{7} = \frac{-10}{35} \][/tex]
Now subtract the fractions:
[tex]\[ (a \times b) - (a \times c) = \frac{-10}{35} - \frac{8}{35} = \frac{-18}{35} \approx -0.5142857142857142 \][/tex]
### Step 6: Verification
At this point, we compare the results from Step 2 and Step 5:
[tex]\[ a \times (b - c) \approx -0.5142857142857142 \][/tex]
[tex]\[ (a \times b) - (a \times c) \approx -0.5142857142857142 \][/tex]
Since both results are approximately equal, we have verified that:
[tex]\[ a \times (b - c) = (a \times b) - (a \times c) \][/tex]
Thus, the equation holds true.
