Let's define the variables to solve the problem systematically:

• Let $x$ be the woman's age today.
• Let $a$, $b$, and $c$ be the ages of her three children today. The two older children are $a$ and $b$, and the youngest is $c$.

Step 1: Set up the conditions 10 years ago.

• 10 years ago, the sum of the ages of the two older children was exactly half of her age. Since their ages today are $a$ and $b$, 10 years ago, their ages were $a - 10$ and $b - 10$, respectively.
• The woman's age 10 years ago was $x - 10$.
• According to the problem, the sum of the ages of the two children 10 years ago was half of the woman's age, so:

$(a - 10) + (b - 10) = \frac{x - 10}{2}$

Step 2: Set up the condition 6 years ago.

• 6 years ago, the woman had her third child. Since the youngest child is $c$ years old today, the youngest child was just born 6 years ago, meaning $c = 6$.

Step 3: The condition for today.

• Today, the sum of the ages of all three children is equal to the woman's current age:

$a + b + c = x$

Since $c = 6$, we can substitute that into the equation:

$a + b + 6 = x$

Step 4: Solve the system of equations.

We now have two equations:

1. From 10 years ago: $(a - 10) + (b - 10) = \frac{x - 10}{2}$ Simplifying: $a + b - 20 = \frac{x - 10}{2}$ $2(a + b - 20) = x - 10$ $2(a + b) - 40 = x - 10$ $2(a + b) = x + 30$

2. From today: $a + b + 6 = x$

Step 5: Substitution and solve for $x$.

Substitute $a + b = x - 6$ into the first equation:

$2(x - 6) = x + 30$ $2x - 12 = x + 30$ $x = 42$

Step 6: Conclusion.

The woman's age today is 42 years.

Would you like more details or have any questions?

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Here are 5 related questions to deepen your understanding:

1. How old are the two older children today?
2. What was the woman’s age when she had her first child?
3. What will the woman's age be when the sum of her children’s ages is double her current age?
4. How would the solution change if the third child were born 8 years ago instead of 6?
5. What algebraic methods can be used to solve similar age-related word problems?

Tip: Always check the reasonableness of your solution by plugging values back into the original conditions to ensure they hold true!