Solving For Apples And Oranges Creating A System Of Equations
Aisha's trip to the market presents us with an interesting mathematical puzzle. She bought a total of 15 fruits, a mix of apples and oranges, spending $9.00 in total. We know the individual prices of each fruit: oranges cost $0.50 each, and apples cost $0.65 each. Our mission is to determine the system of equations that can be used to solve for the number of oranges and apples Aisha purchased. To solve this, we'll use the power of algebra, specifically systems of equations. We'll break down the information into manageable parts, define our variables, and construct equations that represent the given conditions. Let's dive into the world of apples, oranges, and equations!
Defining the Variables The Foundation of Our Equations
The first step in tackling this problem is to clearly define our variables. This is crucial because the variables will serve as the foundation upon which we build our equations. We need symbols to represent the unknowns we are trying to find. In this case, we have two unknowns: the number of oranges and the number of apples Aisha bought. As stated in the problem, let's use '$x$' to represent the number of oranges and '$y$' to represent the number of apples. This simple step of assigning variables transforms the word problem into a language we can manipulate mathematically. With our variables clearly defined, we can now proceed to translate the given information into algebraic equations. Remember, clear variable definitions are the cornerstone of successful problem-solving in algebra. Without them, we would be lost in a sea of unknowns, unable to effectively express the relationships described in the problem. This careful setup is what allows us to move forward with confidence and precision.
Crafting the Equations Translating Words into Math
Now that we have our variables defined, the next crucial step is to translate the information given in the problem into mathematical equations. This is where we bridge the gap between the real-world scenario and the abstract language of algebra. Aisha bought a total of 15 apples and oranges. This information gives us our first equation. We can express the total number of fruits as the sum of the number of oranges (which we've defined as $x$) and the number of apples (which we've defined as $y$). Therefore, our first equation is: $x + y = 15$. This equation captures the quantity aspect of the problem. Next, we need to consider the cost aspect. Aisha spent a total of $9.00. Each orange costs $0.50, and each apple costs $0.65. We can express the total cost as the sum of the cost of the oranges and the cost of the apples. The cost of the oranges is the price per orange ($), which gives us $0.50x$. Similarly, the cost of the apples is the price per apple ($), which gives us $0.65y$. Therefore, our second equation, representing the total cost, is: $0.50x + 0.65y = 9.00$. These two equations together form a system of equations that represents the problem. This system captures both the quantity and the cost constraints of Aisha's purchase, providing a complete mathematical model of the situation. With these equations in hand, we are well-equipped to solve for the unknown quantities.
The System of Equations A Complete Mathematical Picture
By carefully analyzing the information provided in the problem, we've successfully constructed a system of two equations. This system represents the complete mathematical picture of Aisha's purchase. The first equation, $x + y = 15$, tells us about the total number of fruits. It states that the number of oranges ($x$) plus the number of apples ($y$) equals 15. This equation focuses on the quantity aspect of the problem. The second equation, $0.50x + 0.65y = 9.00$, tells us about the total cost. It states that the cost of the oranges (0.50 times the number of oranges) plus the cost of the apples (0.65 times the number of apples) equals $9.00. This equation focuses on the financial aspect of the problem. Together, these two equations form a system that can be used to solve for the two unknowns, $x$ and $y$. This system of equations is a powerful tool because it allows us to consider both constraints (total number of fruits and total cost) simultaneously. Solving this system will give us the specific number of oranges and apples Aisha bought. This is a prime example of how mathematics can be used to model real-world situations and find precise solutions. The system provides a concise and accurate representation of the problem, paving the way for algebraic techniques to be applied.
Solving the System Unveiling the Number of Fruits
While the question asks us to identify the system of equations, it's worthwhile to briefly consider how we might solve this system to find the actual number of oranges and apples. There are several methods we could use, such as substitution or elimination. Let's illustrate the substitution method. From the first equation, $x + y = 15$, we can easily isolate one variable. For example, we can solve for $x$ by subtracting $y$ from both sides, giving us $x = 15 - y$. Now we can substitute this expression for $x$ into the second equation: $0.50x + 0.65y = 9.00$. Replacing $x$ with $15 - y$ gives us $0.50(15 - y) + 0.65y = 9.00$. This equation now has only one variable, $y$, which we can solve for. First, distribute the 0.50: $7.50 - 0.50y + 0.65y = 9.00$. Combine the $y$ terms: $7.50 + 0.15y = 9.00$. Subtract 7.50 from both sides: $0.15y = 1.50$. Finally, divide both sides by 0.15: $y = 10$. So, Aisha bought 10 apples. Now we can substitute this value of $y$ back into either equation to find $x$. Using $x + y = 15$, we get $x + 10 = 15$, which means $x = 5$. Therefore, Aisha bought 5 oranges. This demonstrates how the system of equations we identified can be used to find the specific solution to the problem. While not explicitly required by the question, understanding the solution process highlights the power and utility of setting up the correct system of equations.
Identifying the Correct System The Answer to Our Puzzle
Having walked through the process of defining variables, crafting equations, and even briefly discussing the solution, we are now well-equipped to pinpoint the correct system of equations that represents Aisha's fruit purchase. Recall that we established two key equations. The first equation, $x + y = 15$, represents the total number of fruits, where $x$ is the number of oranges and $y$ is the number of apples. The second equation, $0.50x + 0.65y = 9.00$, represents the total cost, considering the price of each orange ($0.50) and each apple ($0.65). When presented with a set of options, the correct system will be the one that precisely matches these two equations. It's crucial to pay close attention to the coefficients and constants in each equation to ensure they accurately reflect the problem's conditions. For instance, an option that has the equation $0.65x + 0.50y = 9.00$ would be incorrect because it reverses the prices of the apples and oranges. Similarly, an option with an incorrect total number of fruits or an incorrect total cost would also be wrong. The process of systematically building the equations ourselves, as we have done, makes it much easier to identify the correct system among a set of choices. We have a clear understanding of what each equation represents and why it is constructed in a particular way. This solid foundation allows us to confidently select the system that accurately models the given scenario.
Conclusion The Power of Mathematical Modeling
In conclusion, by carefully analyzing the problem, defining variables, and translating the given information into mathematical equations, we have successfully identified the system of equations that can be used to find the number of oranges and apples Aisha bought. The system consists of two equations: $x + y = 15$, representing the total number of fruits, and $0.50x + 0.65y = 9.00$, representing the total cost. This exercise demonstrates the power of mathematical modeling in solving real-world problems. By breaking down a complex scenario into smaller, manageable parts and representing them with algebraic equations, we can gain valuable insights and find precise solutions. The ability to translate words into math is a fundamental skill in problem-solving, and this example illustrates how this skill can be applied in a practical context. Furthermore, understanding the underlying concepts, such as defining variables and constructing equations, allows us to confidently tackle similar problems in the future. The system of equations we've identified provides a complete mathematical representation of Aisha's purchase, allowing us to determine the exact number of each type of fruit she bought. This highlights the elegance and efficiency of using mathematical tools to solve everyday puzzles.
Problem Rewording
Repair Input Keyword: Aisha purchased 15 apples and oranges for $9.00. Each orange costs $0.50, and each apple costs $0.65. If $x$ represents the number of oranges and $y$ represents the number of apples, what system of equations can be used to determine the values of $x$ and $y$?
