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Is it possible for two numbers to have a difference of 8, and a sum of 1?

Let's break down the problem with algebra.

Let the two numbers be represented as \( x \) and \( y \).

We are given two conditions:
1. The difference of the two numbers is 8:  
   \[
   x - y = 8
   \]
2. The sum of the two numbers is 1:  
   \[
   x + y = 1
   \]

Now, we can solve this system of linear equations to find the values of \( x \) and \( y \).

### Step 1: Add the two equations
\[
(x - y) + (x + y) = 8 + 1
\]
Simplifying the left side:
\[
2x = 9
\]
Solving for \( x \):
\[
x = \frac{9}{2} = 4.5
\]

### Step 2: Substitute \( x = 4.5 \) into the second equation
\[
x + y = 1
\]
\[
4.5 + y = 1
\]
Solving for \( y \):
\[
y = 1 - 4.5 = -3.5
\]

### Conclusion:
The two numbers are \( x = 4.5 \) and \( y = -3.5 \). These numbers satisfy both conditions:
- Their difference is \( 4.5 - (-3.5) = 8 \)
- Their sum is \( 4.5 + (-3.5) = 1 \)

So, yes, it is possible for two numbers to have a difference of 8 and a sum of 1.

Would you like to explore further or have any questions about this solution?

### Related Questions:
1. How would you solve the system of equations using substitution or elimination methods?
2. What happens if you change the sum or difference of the numbers in this scenario?
3. What are some real-world situations where such equations might be used?
4. How would the solution change if the numbers were required to be integers?
5. Can the difference and sum be negative, and how does that affect the solutions?

**Tip:** When solving systems of equations, adding the equations is a great way to eliminate one variable, simplifying the process!

Is it possible for two numbers to have a difference of 8, and a sum of 1?

Explore how to determine two numbers whose difference is 8 and sum is 1. This detailed solution includes solving linear equations step-by-step using addition and substitution methods. Ideal for students in Grades 7-10.

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