Let the cost of one apple be $a$ and the cost of one orange be $r$. We can represent the problem with the following two equations:

5a + 4r = 10 \tag{1} 5a + 5r = 11 \tag{2}

Step 1: Eliminate one variable

Subtract equation (1) from equation (2):

$(5a + 5r) - (5a + 4r) = 11 - 10$

Simplify:

$5r - 4r = 1$

$r = 1$

Step 2: Substitute $r = 1$ into one of the original equations

Substitute $r = 1$ into equation (1):

$5a + 4(1) = 10$

Simplify:

$5a + 4 = 10$

$5a = 6$

$a = \frac{6}{5} = 1.20$

Final Answer:

The price of an apple is $1.20, and the price of an orange is $1.00.

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Would you like a detailed step-by-step breakdown of another method, or do you have any questions? Here are related questions for further exploration:

1. How would the solution change if the price for 5 apples and 4 oranges was $12 instead of $10?
2. Can you graphically represent this system of equations to find the solution?
3. What is the significance of solving this problem using substitution instead of elimination?
4. How would the system change if you bought 3 apples and 7 oranges instead?
5. What if there was an error in calculation and one of the totals is rounded? How can we address this?

Tip: When using elimination, always aim to simplify the problem by aligning coefficients for one variable to cancel it out quickly.