To find Lena's profit, we can set up the equation based on her expenses and earnings.
Lena's profit \( P \) can be expressed with the equation:
\[ P = 5n - 200 \]
where:
- \( P \) is the profit,
- \( n \) is the number of pairs of earrings sold,
- \( 5n \) is the total earnings from selling \( n \) pairs of earrings ($5 for each pair),
- \( 200 \) is the booth rental cost.
To find out how many pairs of earrings she must sell to earn a profit of $450, we set \( P \) to 450 and solve for \( n \):
\[ 450 = 5n - 200 \]
Now, add 200 to both sides:
\[ 450 + 200 = 5n \]
\[ 650 = 5n \]
Next, divide by 5:
\[ n = \frac{650}{5} \]
\[ n = 130 \]
Therefore, Lena must sell 130 pairs of earrings to earn a profit of $450.