The problem states that the sum of two rational numbers is [tex]\(-8\)[/tex] and one of the numbers is [tex]\(\frac{-15}{7}\)[/tex]. We need to find the other number. Here’s the step-by-step solution:
1. Identify the total sum and known number:
- The total sum of the two numbers is [tex]\(-8\)[/tex].
- The given number is [tex]\(\frac{-15}{7}\)[/tex].
2. Set up the equation:
Let's denote the unknown number as [tex]\(x\)[/tex].
According to the problem, we have:
[tex]\[
\left( \frac{-15}{7} \right) + x = -8
\][/tex]
3. Isolate [tex]\(x\)[/tex]:
To find [tex]\(x\)[/tex], we move [tex]\(\frac{-15}{7}\)[/tex] to the other side of the equation by adding [tex]\(\frac{15}{7}\)[/tex] to both sides:
[tex]\[
x = -8 - \left( \frac{-15}{7} \right)
\][/tex]
4. Combine the terms:
Convert [tex]\(-8\)[/tex] to a fraction with a common denominator to simplify the calculation:
[tex]\[
-8 = \frac{-8 \times 7}{7} = \frac{-56}{7}
\][/tex]
Now we can write the equation as:
[tex]\[
x = \frac{-56}{7} + \frac{15}{7}
\][/tex]
5. Simplify the fractions:
Since the denominators are the same, we can combine the numerators:
[tex]\[
x = \frac{-56 + 15}{7} = \frac{-41}{7}
\][/tex]
6. Convert to a decimal (if necessary):
The fraction [tex]\(\frac{-41}{7}\)[/tex] can be converted to a decimal for clarity:
[tex]\[
x = -5.857142857142857
\][/tex]
Therefore, the other number is [tex]\(\frac{-41}{7}\)[/tex] or approximately [tex]\(-5.857142857142857\)[/tex].
Recap:
- One number is [tex]\(\frac{-15}{7}\)[/tex].
- The other number is [tex]\(\frac{-41}{7}\)[/tex] or [tex]\(-5.857142857142857\)[/tex].
- When these numbers are added, their sum is [tex]\(-8\)[/tex].
