Age Relationship Problem: Solve for Hector's Age Using 3x and -10 Relations

Q: Which person's age should I use as my variable x?

Choose the person that others are compared to. Since Luciana is "3 times Ivana's age" and Hector is compared to Luciana, make Ivana's age = x. This keeps your expressions simple!

Q: How do I translate "10 years younger than Luciana"?

If Luciana is $ 3x $ years old, then someone 10 years younger is $ 3x - 10 $. Remember: younger means subtract, older means add!

Q: How can I check if my final answer makes sense?

Verify two things: (1) All ages are positive, and (2) The relationships work. Here: Ivana = 6, Luciana = 18 (which is 3×6 ✓), Hector = 8 (which is 18-10 ✓).

Q: Why do we combine like terms to get 7x?

In the equation $ x + 3x + (3x - 10) = 32 $, we have three x-terms: x + 3x + 3x = 7x. Always collect all x-terms together before solving!

System of Age Relationships with Linear Variables

Luciana's age is 3 times Ivana's age.

Hector is 10 years younger than Luciana.

The sum of the three ages is 32.

How old is Hector?

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Luciana's age is 3 times Ivana's age.

Hector is 10 years younger than Luciana.

The sum of the three ages is 32.

How old is Hector?

Step-by-step solution

To solve this problem, let's define the variables and set up the equation:

Let Ivana's age be $x$ .
Then, Luciana's age is $3x$ , since her age is three times Ivana's age.
Hector, being 10 years younger than Luciana, would be $3x - 10$ .

We know the sum of their ages is 32, so we set up the equation:

$x + 3x + (3x - 10) = 32$

Simplify the equation:

$x + 3x + 3x - 10 = 32$

$7x - 10 = 32$

Next, solve for $x$ :

Add 10 to both sides:

$7x = 42$

Divide both sides by 7:

$x = 6$

Now that we know Ivana is 6 years old, we can find Hector's age:

Hector's age = $3x - 10 = 3(6) - 10 = 18 - 10 = 8$

Therefore, Hector is 8 years old.

Final Answer

$8$

Key Points to Remember

Essential concepts to master this topic

Variable Setup: Let one person's age be x, express others in terms of x
Translation: "3 times" becomes 3x, "10 years younger" becomes 3x - 10
Verification: Check that all three ages sum to 32: 6 + 18 + 8 = 32 ✓

Common Mistakes

Avoid these frequent errors

Setting up variables incorrectly for age relationships
Don't make Hector's age x and then struggle with complex expressions = confusing algebra! This makes the "3 times" and "10 years younger" relationships much harder to write. Always choose the simplest person (usually the one others are compared to) as your variable x.

Practice Quiz

Test your knowledge with interactive questions

$$ x+7=14 $$

$x=\text{?}$

FAQ

Everything you need to know about this question

Which person's age should I use as my variable x?

Choose the person that others are compared to. Since Luciana is "3 times Ivana's age" and Hector is compared to Luciana, make Ivana's age = x. This keeps your expressions simple!

How do I translate "10 years younger than Luciana"?

If Luciana is $3x$ years old, then someone 10 years younger is $3x - 10$ . Remember: younger means subtract, older means add!

What if I get a negative age in my calculations?

Check your setup! Negative ages usually mean you set up the relationships backwards. Make sure "younger than" uses subtraction and "older than" uses addition.

How can I check if my final answer makes sense?

Verify two things: (1) All ages are positive, and (2) The relationships work. Here: Ivana = 6, Luciana = 18 (which is 3×6 ✓), Hector = 8 (which is 18-10 ✓).

Why do we combine like terms to get 7x?

In the equation $x + 3x + (3x - 10) = 32$ , we have three x-terms: x + 3x + 3x = 7x. Always collect all x-terms together before solving!

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