4. Application: Word Problems

Convert real-life situations into equations.

In this chunk, we learn how to translate real-life scenarios into mathematical equations, specifically for simultaneous equations. This means we take information from a situation, like the prices of tickets or quantities, and create equations that capture those details. Understanding how to express real situations as mathematical equations is a key skill in algebra, as it allows us to solve practical problems more effectively.

Imagine you're planning a party and need to figure out the total cost of snacks. You find out that chips cost $3 per bag and drinks are$2 each. If you buy 4 bags of chips and 6 drinks, you want to know how much money you'll spend in total. By creating an equation based on the prices and quantities, you can calculate the total cost efficiently.

Example: A cinema charges $8 for a child and$12 for an adult. If 5 children and 3 adults together cost $84, and 2 children and 5 adults cost$86, find the price of each ticket.

Here, we are given specific prices and quantities related to children's and adults' cinema tickets. The information is formatted into two equations: the first equation represents the total cost of 5 children and 3 adults, while the second represents the total cost of 2 children and 5 adults. By setting up these equations, we prepare to solve the problem systematically using either substitution or elimination methods.

Think of this situation like a local market where you want to buy fruits. You know that apples are $2 each and bananas are$1 each. If you buy 3 apples and 2 bananas, the total cost should be $10 for the first scenario. The second scenario could be if you buy 1 apple and 4 bananas for a total of$8. This setup helps you figure out how much each fruit costs based on your purchases.

Let x = cost of child ticket, y = cost of adult ticket. Equations: 5x + 3y = 84, 2x + 5y = 86. Solve using elimination or substitution.

In this step, we define our variables: x for the cost of a child ticket and y for the cost of an adult ticket. The two equations we formed from the earlier situation are ready to be manipulated and solved for x and y. Using methods like elimination (where we align and subtract equations) or substitution (replacing one variable with the other) allows us to isolate and solve for the ticket prices.

Imagine you have a puzzle to complete, where each piece represents an equation. To solve it, you need to find how the pieces fit together. By substituting one piece for another in your calculations (like substituting values), you can start seeing the bigger picture—the price of the tickets—and successfully complete your puzzle.

Multiply first by 5, second by 3: 25x + 15y = 420, 6x + 15y = 258. Subtract: 19x = 162 ⇒ x = 8.53. Then find y.

After setting up our equations, we manipulate them by multiplying to align the coefficients of y, which simplifies the subtraction process. Once we compute the value of x, we backtrack to find the value of y. This systematic approach of solving and substituting helps ensure we have accurate results for each ticket's price.

Imagine baking cookies, where you need to adjust the amount of flour (x) and sugar (y) based on your recipe. By substituting values and adjusting quantities, you can perfectly balance your ingredients to end up with delicious cookies, just as we adjust our equations to find the right prices.

Answer: Child = $8.53, solve for Adult.

In the conclusion, we summarize the solution—finding the cost of each ticket. This step reinforces the importance of checking our work and ensuring that both equations satisfy our final values for x and y. Understanding how to interpret our results is crucial when applying these concepts to real-world situations.

Think of this like finalizing your budget after shopping. After calculating the cost of items in your cart, you want to ensure everything matches your budget. By reviewing each item’s price, just like we check each variable in our equations, you can be confident that your financial plan is accurate.

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