To solve the problem of determining the current ages of María, Alejandra, and Fiorella (denoted as [tex]\(a\)[/tex] and [tex]\(b\)[/tex] respectively), given certain relationships between their ages, follow these steps:
1. Set Up the Equations Based on the Given Age Relationships:
We are given two relationships for the ages of María and Fiorella:
- Three years ago:
- María's age: [tex]\(a - 3\)[/tex]
- Fiorella's age: [tex]\(b - 3\)[/tex]
The ages were in the ratio [tex]\(4:7\)[/tex], so we have:
[tex]\[
\frac{a - 3}{b - 3} = \frac{4}{7}
\][/tex]
- Two years from now:
- María's age: [tex]\(a + 2\)[/tex]
- Fiorella's age: [tex]\(b + 2\)[/tex]
The ages will be in the ratio [tex]\(3:5\)[/tex], so we have:
[tex]\[
\frac{a + 2}{b + 2} = \frac{3}{5}
\][/tex]
2. Translate the Ratios into Equations:
From the first age relationship:
[tex]\[
7(a - 3) = 4(b - 3)
\][/tex]
Simplifying:
[tex]\[
7a - 21 = 4b - 12
\][/tex]
Rearrange the equation:
[tex]\[
7a - 4b = 9 \quad \text{(Equation 1)}
\][/tex]
From the second age relationship:
[tex]\[
5(a + 2) = 3(b + 2)
\][/tex]
Simplifying:
[tex]\[
5a + 10 = 3b + 6
\][/tex]
Rearrange the equation:
[tex]\[
5a - 3b = -4 \quad \text{(Equation 2)}
\][/tex]
3. Solve the System of Linear Equations:
We now have a system of two linear equations:
[tex]\[
\begin{cases}
7a - 4b = 9 & \text{(Equation 1)}\\
5a - 3b = -4 & \text{(Equation 2)}
\end{cases}
\][/tex]
To solve this system, we use the method of elimination or substitution. Let's use the elimination method for clarity:
First, we multiply Equation 1 by 3 and Equation 2 by 4 to align the coefficients of [tex]\(b\)[/tex]:
[tex]\[
3(7a - 4b) = 3 \cdot 9 \implies 21a - 12b = 27
\][/tex]
[tex]\[
4(5a - 3b) = 4 \cdot (-4) \implies 20a - 12b = -16
\][/tex]
Now subtract the second resulting equation from the first:
[tex]\[
(21a - 12b) - (20a - 12b) = 27 - (-16)
\][/tex]
Simplifying this:
[tex]\[
a = 43
\][/tex]
Substitute [tex]\( a = 43 \)[/tex] back into one of the original equations to find [tex]\( b \)[/tex]. Using Equation 1:
[tex]\[
7(43) - 4b = 9
\][/tex]
[tex]\[
301 - 4b = 9
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
301 - 9 = 4b
\][/tex]
[tex]\[
292 = 4b
\][/tex]
[tex]\[
b = 73
\][/tex]
4. Calculate the Sum of the Ages:
[tex]\[
a + b = 43 + 73 = 116
\][/tex]
Therefore, the sum of María and Fiorella's current ages is [tex]\(\boxed{116}\)[/tex].
