Certainly! Let’s solve this step-by-step to find the pair of integers whose sum is -13 and whose difference is -7.
1. Denote the two integers as [tex]\( a \)[/tex] and [tex]\( b \)[/tex].
2. According to the problem, we have two main conditions:
- The sum of the integers is [tex]\( a + b = -13 \)[/tex].
- The difference of the integers is [tex]\( a - b = -7 \)[/tex].
3. To solve for the integers, we can use these two equations:
[tex]\[ a + b = -13 \][/tex]
[tex]\[ a - b = -7 \][/tex]
4. First, add the two equations together to eliminate [tex]\( b \)[/tex]:
[tex]\[ (a + b) + (a - b) = -13 - 7 \][/tex]
[tex]\[ a + b + a - b = -20 \][/tex]
Simplifying, we get:
[tex]\[ 2a = -20 \][/tex]
5. Solve for [tex]\( a \)[/tex]:
[tex]\[ a = \frac{-20}{2} \][/tex]
[tex]\[ a = -10 \][/tex]
6. Substitute [tex]\( a = -10 \)[/tex] back into the first equation to find [tex]\( b \)[/tex]:
[tex]\[ a + b = -13 \][/tex]
[tex]\[ -10 + b = -13 \][/tex]
7. Solve for [tex]\( b \)[/tex]:
[tex]\[ b = -13 + 10 \][/tex]
[tex]\[ b = -3 \][/tex]
Therefore, the pair of integers is [tex]\( \boxed{(-10, -3)} \)[/tex].
To verify, their sum is:
[tex]\[ -10 + (-3) = -13 \][/tex]
And their difference is:
[tex]\[ -10 - (-3) = -10 + 3 = -7 \][/tex]
This confirms that the pair [tex]\( (-10, -3) \)[/tex] satisfies both conditions given in the problem.
