Find Two Ages Using a System of Equations

What This Problem Teaches

• How to translate age relationships into algebraic expressions
• Setting up a single-variable equation from a system of constraints
• Working with the phrase "older than twice" in mathematical context
• Verifying solutions by checking both the sum and the relationship
• Recognizing when substitution simplifies a two-variable system into one equation

Solution: Method 1 — The Substitution Strategy

When one age is defined in terms of another, we can use a single variable and substitute to create one equation.

Step 1 — Choose the variable

Let t = Teo's age in years. We choose Teo because Richard's age is described in terms of Teo's age.

Step 2 — Express Richard's age

The phrase "Richard is 3 years older than twice Teo's age" translates to:

This means we first take twice Teo's age (2t), then add 3 years to get Richard's age.

Step 3 — Set up the sum equation

Their combined age is 24, so:

Step 4 — Solve for Teo's age

Combine like terms on the left side:

So Teo is 7 years old.

Step 5 — Find Richard's age

Substitute t = 7 into Richard's age formula:

So Richard is 17 years old.

Solution: Method 2 — The Two-Variable System

We can also solve this using two variables and formal substitution between equations.

Step 1 — Define both variables

Let R = Richard's age and T = Teo's age.

Step 2 — Write both equations

From "combined age of 24":

From "Richard is 3 years older than twice Teo's age":

Step 3 — Substitute equation 2 into equation 1

Replace R in equation 1 with 2T + 3:

Step 4 — Find Richard's age using equation 2

Both methods give us the same answer: Teo is 7 and Richard is 17.

Verification

Let's check both conditions with our answer:

Check the sum

Check the relationship

Both conditions are satisfied, confirming our answer is correct.

Common Pitfalls

How to Spot This Problem Type

Look for these key phrases in age problems:

• "Combined age" or "total age" — signals you need a sum equation
• "older than twice" or "younger than double" — indicates a linear relationship
• "3 years older than twice [name's] age" — translates to 2x + 3
• Two people with one constraint each — perfect for substitution method

The Underlying Pattern

This problem follows the classic sum-and-relationship pattern:

In our case: S = 24, m = 2, b = 3, so x = (24 - 3)/(2 + 1) = 21/3 = 7.

This formula works whenever one quantity is a linear function of the other. Age problems, money problems, and measurement problems often follow this structure.