The present age of a woman is 3 years more than three times the age of her daughter. Three years hence, the woman's age will be 10 years, more than twice the age of her daughter. Find their present ages.

Explanation

To solve the problem of finding the present ages of the woman and her daughter, we can follow these steps:

### Step 1: Define the Variables
Let:
- \( x \) = present age of the woman
- \( y \) = present age of the daughter

### Step 2: Set Up the First Equation
According to the problem, the present age of the woman is 3 years more than three times the age of her daughter. This can be expressed as:
\[ x = 3 + 3y \]
Rearranging this gives us:
\[ x - 3y = 3 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
The problem also states that three years hence (in 3 years), the woman's age will be 10 years more than twice the age of her daughter. In 3 years, their ages will be \( x + 3 \) and \( y + 3 \), respectively. This gives us the equation:
\[ x + 3 = 10 + 2(y + 3) \]
Simplifying this:
\[ x + 3 = 10 + 2y + 6 \]
\[ x + 3 = 16 + 2y \]
Rearranging gives:
\[ x - 2y = 16 - 3 \]
\[ x - 2y = 13 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
Now we have a system of two equations:
1. \( x - 3y = 3 \)
2. \( x - 2y = 13 \)

We can subtract Equation 1 from Equation 2:
\[ (x - 2y) - (x - 3y) = 13 - 3 \]
This simplifies to:
\[ -y = 10 \]
Thus, we find:
\[ y = -10 \quad \text{(This is incorrect; let's correct the subtraction)} \]

Instead, let's subtract correctly:
\[ x - 2y - (x - 3y) = 13 - 3 \]
This simplifies to:
\[ y = 10 \]

### Step 5: Find the Woman's Age
Now that we have \( y = 10 \), we can substitute this value back into Equation 1 to find \( x \):
\[ x - 3(10) = 3 \]
\[ x - 30 = 3 \]
\[ x = 33 \]

### Conclusion
The present ages are:
- The woman's age \( x = 33 \) years
- The daughter's age \( y = 10 \) years