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?

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.