If three numbers are added in pairs, the sums equal 10, 19 and 21. Find the numbers.

Options

A. 6, 4, 15
B. 1, 9, 12
C. 9, 10, 2
D. 5, 6, 10

Answer

To solve the problem of finding three numbers \( x \), \( y \), and \( z \) given that their pairwise sums are 10, 19, and 21, we can follow these steps:

### Step 1: Set up the equations based on the given sums.
We know:
1. \( x + y = 10 \) (Equation 1)
2. \( y + z = 19 \) (Equation 2)
3. \( z + x = 21 \) (Equation 3)

### Step 2: Add all three equations together.
Adding all three equations gives us:
\[ (x + y) + (y + z) + (z + x) = 10 + 19 + 21 \]
This simplifies to:
\[ 2x + 2y + 2z = 50 \]

### Step 3: Simplify the equation.
Dividing the entire equation by 2, we get:
\[ x + y + z = 25 \quad \text{(Equation 4)} \]

### Step 4: Use Equation 4 to find each variable.
Now we can use Equation 4 along with the individual equations to find \( x \), \( y \), and \( z \).

#### Step 4.1: Find \( z \) using Equation 1.
From Equation 1:
\[ y = 10 - x \]
Substituting \( y \) into Equation 4:
\[ x + (10 - x) + z = 25 \]
This simplifies to:
\[ 10 + z = 25 \implies z = 25 - 10 = 15 \]

#### Step 4.2: Find \( y \) using Equation 2.
Now substituting \( z = 15 \) into Equation 2:
\[ y + 15 = 19 \implies y = 19 - 15 = 4 \]

#### Step 4.3: Find \( x \) using Equation 1.
Now substituting \( y = 4 \) into Equation 1:
\[ x + 4 = 10 \implies x = 10 - 4 = 6 \]

### Final Result:
The three numbers are:
\[ x = 6, \quad y = 4, \quad z = 15 \]

### Summary:
The numbers are \( x = 6 \), \( y = 4 \), and \( z = 15 \).