Question

1. twice a number added to another number is -8. the difference of the two numbers is 2. find the numbers.

Explanation:

Step1: Define variables

Let the first number be \( x \) and the second number be \( y \).

Step2: Set up equations

From "Twice a number added to another number is -8", we get \( 2x + y = -8 \).
From "The difference of the two numbers is 2", we get \( x - y = 2 \) (assuming \( x>y \), we could also have \( y - x = 2 \), but we'll solve this first).

Step3: Solve the system of equations

We can use the elimination method. Add the two equations:
\( (2x + y) + (x - y) = -8 + 2 \)
Simplify: \( 3x = -6 \)

Step4: Find \( x \)

Divide both sides by 3: \( x = \frac{-6}{3} = -2 \)

Step5: Find \( y \)

Substitute \( x = -2 \) into \( x - y = 2 \): \( -2 - y = 2 \)
Add \( y \) to both sides: \( -2 = 2 + y \)
Subtract 2 from both sides: \( y = -2 - 2 = -4 \)
Let's check with the first equation: \( 2(-2) + (-4) = -4 - 4 = -8 \), which matches. If we had taken \( y - x = 2 \), let's see:
Equations: \( 2x + y = -8 \) and \( y - x = 2 \) (so \( y = x + 2 \))
Substitute \( y = x + 2 \) into \( 2x + y = -8 \): \( 2x + x + 2 = -8 \)
\( 3x = -10 \), \( x = -\frac{10}{3} \), then \( y = -\frac{10}{3} + 2 = -\frac{4}{3} \)
Check: \( 2(-\frac{10}{3}) + (-\frac{4}{3}) = -\frac{20}{3} - \frac{4}{3} = -\frac{24}{3} = -8 \), which also matches. But the problem says "the numbers", probably expecting integer solutions, so the first solution with \( x = -2 \), \( y = -4 \) is more likely.

Answer:

The two numbers are \( -2 \) and \( -4 \) (or if we consider the other case, \( -\frac{10}{3} \) and \( -\frac{4}{3} \), but the integer solution is more probable here).