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To solve this problem, let's define the two unknown numbers and set up equations based on the information provided.

### Step 1: Define the Variables
Let:
- \( x \) = the first number
- \( y \) = the second number

### Step 2: Set Up Equations Based on the Problem
1. The sum of the two numbers is seventy-two:
   \[
   x + y = 72
   \]
   
2. Two times the first number equals ten times the second number:
   \[
   2x = 10y
   \]

### Step 3: Solve the Equations

#### Equation 2 can be simplified:
\[
2x = 10y
\]
Divide both sides by 2:
\[
x = 5y
\]

#### Substitute \( x = 5y \) into Equation 1:
\[
5y + y = 72
\]
Combine like terms:
\[
6y = 72
\]
Divide by 6:
\[
y = 12
\]

#### Substitute \( y = 12 \) back into \( x = 5y \):
\[
x = 5 \times 12 = 60
\]

### Solution
The two numbers are:
- First number, \( x = 60 \)
- Second number, \( y = 12 \)

### Verification
1. The sum of the numbers: \( 60 + 12 = 72 \), which matches the problem's condition.
2. Two times the first number: \( 2 \times 60 = 120 \) and ten times the second number: \( 10 \times 12 = 120 \), so this condition is also satisfied.

### Answer
The numbers are **60 and 12**.

Would you like further explanation on any of these steps?

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Here are some additional questions related to this problem:

1. How would the solution change if the sum was different, say, 80 instead of 72?
2. What if the relationship between the numbers was "three times the first number equals fifteen times the second"?
3. Can you find a general formula to solve for two numbers if given their sum and a relationship between them?
4. How could this be solved using a graph?
5. Could you use substitution or elimination methods to solve other types of linear systems?

**Tip:** Setting up equations based on given conditions is crucial in word problems—always start by defining your variables and interpreting the conditions step-by-step.

The sum of two numbers is seventy-two. Two times the first number equals ten times the second number. What are the numbers?

Learn how to solve a problem involving two unknown numbers with a given sum of 72 and a relationship where twice the first number equals ten times the second. This example demonstrates basic algebraic concepts and the substitution method, suitable for Grades 6-8.

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