Let's solve the problem step-by-step.
We are given two pieces of information:
1. Lian and her younger sister Ella are 4 years apart.
2. The sum of their ages is 20.
We are to find the ages of Lian and Ella. Let's define:
- [tex]\( l \)[/tex] as Lian's age in years.
- [tex]\( e \)[/tex] as Ella's age in years.
We can form the following system of linear equations:
1. [tex]\( l - 4 = e \)[/tex] (since Lian is 4 years older than Ella)
2. [tex]\( l + e = 20 \)[/tex] (since the sum of their ages is 20)
Now we'll solve this system of equations step-by-step.
### Step 1: Express [tex]\( e \)[/tex] in terms of [tex]\( l \)[/tex]
From the first equation:
[tex]\[ l - 4 = e \][/tex]
### Step 2: Substitute [tex]\( e \)[/tex] in the second equation
Substitute [tex]\( e \)[/tex] from the first equation into the second equation:
[tex]\[ l + (l - 4) = 20 \][/tex]
### Step 3: Combine like terms
[tex]\[ l + l - 4 = 20 \][/tex]
[tex]\[ 2l - 4 = 20 \][/tex]
### Step 4: Solve for [tex]\( l \)[/tex]
Add 4 to both sides:
[tex]\[ 2l = 24 \][/tex]
Divide by 2:
[tex]\[ l = 12 \][/tex]
So, Lian is 12 years old.
### Step 5: Find Ella's age
Using the first equation [tex]\( e = l - 4 \)[/tex]:
[tex]\[ e = 12 - 4 \][/tex]
[tex]\[ e = 8 \][/tex]
So, Ella is 8 years old.
Thus, Lian is [tex]\( \boxed{12} \)[/tex] years old and Ella is [tex]\( \boxed{8} \)[/tex] years old.
