To determine how much money Alex had initially, let’s go through the problem step by step.
### Step 1: Define the Variables
Let [tex]\( T \)[/tex], [tex]\( N \)[/tex], and [tex]\( A \)[/tex] be the initial amounts of money Tierra, Nico, and Alex had, respectively. Let [tex]\( x \)[/tex] be the amount of money they each had left after spending.
### Step 2: Set Up the Equations
We know the total amount of money they had initially:
[tex]\[ T + N + A = 865 \][/tex]
We are given information about how much each person spent and how much they had left:
- Tierra spent [tex]\(\frac{2}{5}\)[/tex] of her money:
[tex]\[ T_{left} = T - \frac{2}{5}T = \frac{3}{5}T \][/tex]
- Nico spent [tex]\( \$40 \)[/tex]:
[tex]\[ N_{left} = N - 40 \][/tex]
- Alex spent twice as much as Tierra:
[tex]\[ A_{spent} = 2 \times \frac{2}{5}T = \frac{4}{5}T \][/tex]
So, Alex’s remaining money is:
[tex]\[ A_{left} = A - \frac{4}{5}T \][/tex]
Since after spending, they all had the same amount of money left:
[tex]\[ \frac{3}{5}T = N - 40 = A - \frac{4}{5}T \][/tex]
### Step 3: Solve the System of Equations
We now have three key equations:
[tex]\[ T + N + A = 865 \quad \text{(Equation 1)} \][/tex]
[tex]\[ \frac{3}{5}T = N - 40 \quad \text{(Equation 2)} \][/tex]
[tex]\[ \frac{3}{5}T = A - \frac{4}{5}T \quad \text{(Equation 3)} \][/tex]
#### Simplify Equation 3:
[tex]\[ \frac{3}{5}T + \frac{4}{5}T = A \][/tex]
[tex]\[ \frac{7}{5}T = A \][/tex]
[tex]\[ A = \frac{7}{5}T \quad \text{(Simplified form of Equation 3)} \][/tex]
#### Use Equation 2:
[tex]\[ N = \frac{3}{5}T + 40 \quad \text{(Rearranged form of Equation 2)} \][/tex]
#### Substitute [tex]\(N\)[/tex] and [tex]\(A\)[/tex] back into Equation 1:
[tex]\[ T + \left(\frac{3}{5}T + 40\right) + \frac{7}{5}T = 865 \][/tex]
[tex]\[ T + \frac{3}{5}T + 40 + \frac{7}{5}T = 865 \][/tex]
Combine like terms:
[tex]\[ T + \frac{3}{5}T + \frac{7}{5}T + 40 = 865 \][/tex]
[tex]\[ T + 2T + 40 = 865 \][/tex]
[tex]\[ 3T + 40 = 865 \][/tex]
Solve for [tex]\(T\)[/tex]:
[tex]\[ 3T = 825 \][/tex]
[tex]\[ T = 275 \][/tex]
Now use the value of [tex]\(T\)[/tex] to find [tex]\(N\)[/tex] and [tex]\(A\)[/tex]:
[tex]\[ N = \frac{3}{5} \times 275 + 40 \][/tex]
[tex]\[ N = 165 + 40 \][/tex]
[tex]\[ N = 205 \][/tex]
[tex]\[ A = \frac{7}{5} \times 275 \][/tex]
[tex]\[ A = 385 \][/tex]
### Conclusion
Therefore, Alex initially had:
[tex]\[ \boxed{385} \][/tex]
So, Alex had [tex]\( \$385 \)[/tex] in the beginning.
