Let \( x \) be the amount invested at 5%, and \( y \) be the amount invested at 6%. According to the problem, we have the following two equations:
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The total investment is $10,000: \[ x + y = 10000 \]
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The total annual income from both investments is $550: \[ 0.05x + 0.06y = 550 \]
Now, we can solve these equations step by step.
Step 1: Solve for \( y \) in terms of \( x \) using the first equation: \[ y = 10000 - x \]
Step 2: Substitute \( y \) into the second equation: \[ 0.05x + 0.06(10000 - x) = 550 \]
Step 3: Expand and simplify the equation: \[ 0.05x + 600 - 0.06x = 550 \] \[ 600 - 0.01x = 550 \]
Step 4: Solve for \( x \): \[ 600 - 550 = 0.01x \] \[ 50 = 0.01x \] \[ x = 50 / 0.01 \] \[ x = 5000 \]
Step 5: Calculate \( y \): Using \( y = 10000 - x \): \[ y = 10000 - 5000 = 5000 \]
Final Answer: Esraa invested $5,000 at 5% and $5,000 at 6%.
Verification:
- Income from 5% investment: \( 0.05 \times 5000 = 250 \)
- Income from 6% investment: \( 0.06 \times 5000 = 300 \)
- Total income: \( 250 + 300 = 550 \) (correct)
Thus, the amounts are verified. Esraa invested $5,000 at 5% and $5,000 at 6%.