Question

1. there are three integers. the sums of each distinct pair of integers are 6, 8, and 18. what is the greatest integer?

Explanation:

Step1: Define variables for the integers

Let the three integers be \( x \), \( y \), and \( z \). Then we have the following equations based on the sums of each distinct pair:
\( x + y = 6 \)
\( x + z = 8 \)
\( y + z = 18 \)

Step2: Add all three equations together

Adding the three equations: \( (x + y) + (x + z) + (y + z) = 6 + 8 + 18 \)
Simplify the left - hand side: \( 2x + 2y + 2z=32 \)
Divide both sides by 2: \( x + y + z = 16 \)

Step3: Solve for each variable

We know that \( x + y = 6 \). Substitute this into \( x + y + z = 16 \):
\( 6+z = 16 \), then \( z=16 - 6=10 \)
We know that \( x + z = 8 \). Substitute \( z = 10 \) into it: \( x+10 = 8 \), then \( x=8 - 10=- 2 \)
We know that \( y + z = 18 \). Substitute \( z = 10 \) into it: \( y + 10=18 \), then \( y = 18 - 10 = 8 \)

Answer:

The greatest integer among - 2, 8, and 10 is 10. So the answer is 10.