Let's solve the problem step by step:
### Step 1: Define the Equations
We need to find two integers, \(a\) and \(b\), such that:
1. Their sum is \(-3\):
[tex]\[ a + b = -3 \][/tex]
2. Their difference is \(-5\):
[tex]\[ a - b = -5 \][/tex]
### Step 2: Solve the System of Equations
We have the following system of linear equations from the given conditions:
1. \( a + b = -3 \)
2. \( a - b = -5 \)
First, let's add these two equations together:
[tex]\[ (a + b) + (a - b) = -3 + (-5) \][/tex]
This simplifies to:
[tex]\[ 2a = -8 \][/tex]
[tex]\[ a = -4 \][/tex]
### Step 3: Substitute the Value of \(a\)
Now, we take the value of \(a\) and substitute it back into the first equation (\(a + b = -3\)):
[tex]\[ -4 + b = -3 \][/tex]
[tex]\[ b = -3 + 4 \][/tex]
[tex]\[ b = 1 \][/tex]
### Step 4: Verify the Solution
We found two integers, \(a = -4\) and \(b = 1\). Let's verify if they satisfy both conditions:
1. Sum is \(-3\):
[tex]\[ -4 + 1 = -3 \][/tex]
2. Difference is \(-5\):
[tex]\[ -4 - 1 = -5 \][/tex]
### Step 5: Address the Additional Condition
While the third condition (\(a - b = 2\)) appears in the text, we don't need it to solve the primary conditions of sum and difference provided. The correct integers satisfying the sum \(-3\) and difference \(-5\) are \(a = -4\) and \(b = 1\).
The pair of integers meeting the given conditions are:
[tex]\[ (-4, 1) \][/tex]
