Solving Word Problems With Systems Of Equations

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Hey guys! Let's dive into the fascinating world of solving word problems using systems of equations. It might sound intimidating at first, but trust me, it's like cracking a code – super satisfying when you get it! We'll break down the process step-by-step, making it easy to understand and apply. This method is incredibly useful for tackling various real-life situations, from calculating costs to determining mixtures. So, let's jump right in and explore how to translate those tricky word problems into solvable math problems. Get ready to boost your problem-solving skills!

Problem: Books and Notebooks

Let's start with a classic example: "Two identical books and three identical notebooks cost 29 zlotys. The price of a notebook was 0.3 of the price of a book. Calculate the price of the book." This is a perfect scenario to apply our systems of equations knowledge. The key here is to identify the unknowns and the relationships between them. We have the price of the book and the price of the notebook as our unknowns. The problem gives us two crucial pieces of information: the total cost and the relationship between the prices. This information will help us form our equations. So, let's break it down and see how we can transform these words into mathematical expressions. Remember, the goal is to create a clear and solvable system that represents the problem accurately. By carefully analyzing the problem, we can set the stage for a smooth and logical solution.

Setting up the Equations

Okay, first things first, let's define our variables. Let's say the price of one book is represented by 'x' and the price of one notebook is represented by 'y'. Now, we can translate the given information into equations. "Two identical books and three identical notebooks cost 29 zlotys" translates to: 2x + 3y = 29. This is our first equation, representing the total cost. Next, we have "The price of a notebook was 0.3 of the price of a book," which translates to: y = 0.3x. This is our second equation, showing the relationship between the prices of the notebook and the book. Now, we have a system of two equations with two variables:

  • 2x + 3y = 29
  • y = 0.3x

This is the heart of the problem! We've successfully converted the word problem into a mathematical system. The next step is to solve this system. We have a couple of methods we can use, such as substitution or elimination. In this case, since we already have 'y' expressed in terms of 'x' in the second equation, the substitution method might be the most straightforward approach. So, let's dive into solving this system and find the values of 'x' and 'y'.

Solving the System by Substitution

Alright, let's tackle this system of equations using the substitution method. We know from our second equation that y = 0.3x. So, we can substitute this expression for 'y' into our first equation. This means we'll replace 'y' in the equation '2x + 3y = 29' with '0.3x'. Doing that, we get:

  • 2x + 3(0.3x) = 29

Now, we have an equation with only one variable, 'x'. Let's simplify and solve for 'x'. First, multiply 3 by 0.3x, which gives us 0.9x. Our equation now looks like this:

  • 2x + 0.9x = 29

Next, combine the 'x' terms. 2x plus 0.9x equals 2.9x. So, we have:

  • 2.9x = 29

To isolate 'x', we need to divide both sides of the equation by 2.9:

  • x = 29 / 2.9

Calculating this, we find that:

  • x = 10

So, we've found that the price of one book (x) is 10 zlotys! That's a big step. Now that we know the value of 'x', we can easily find the value of 'y' using our second equation (y = 0.3x). Let's do that in the next step.

Finding the Price of the Notebook

Awesome, we've already figured out that the price of one book (x) is 10 zlotys. Now, let's find the price of the notebook (y). Remember, we have the equation y = 0.3x. To find 'y', we simply substitute the value of 'x' (which is 10) into this equation. So, we get:

  • y = 0.3 * 10

Multiplying 0.3 by 10, we get:

  • y = 3

Therefore, the price of one notebook (y) is 3 zlotys. We've successfully found both the price of the book and the price of the notebook. But, it's always a good idea to check our answers to make sure they make sense in the context of the original problem. Let's do that in the next section to ensure we've got the correct solution!

Checking the Solution

Okay, we've found that the price of a book (x) is 10 zlotys and the price of a notebook (y) is 3 zlotys. Now, the crucial step: let's check if these values satisfy the conditions of our original problem. Remember, the problem stated that "Two identical books and three identical notebooks cost 29 zlotys." So, let's plug our values into this condition:

  • 2 * (price of a book) + 3 * (price of a notebook) = 29
  • 2 * (10) + 3 * (3) = 29
  • 20 + 9 = 29
  • 29 = 29

Great! The first condition is satisfied. Now, let's check the second condition: "The price of a notebook was 0.3 of the price of a book." We found that the notebook costs 3 zlotys and the book costs 10 zlotys. Let's see if 3 is indeed 0.3 times 10:

    1. 3 * (price of a book) = price of a notebook
    1. 3 * (10) = 3
  • 3 = 3

Fantastic! The second condition is also satisfied. This means our solution is correct. We've successfully checked our work and confirmed that a book costs 10 zlotys and a notebook costs 3 zlotys. This step is super important to avoid mistakes and ensure accuracy in your problem-solving. Now that we've got the right answer, let's state our final answer clearly.

Stating the Final Answer

Alright, we've gone through the whole process – setting up equations, solving the system, and checking our solution. Now, it's time to state our final answer clearly and concisely. The problem asked us to calculate the price of the book. We found that the price of the book (x) is 10 zlotys. So, our final answer is:

The price of the book is 10 zlotys.

See? That wasn't so bad, was it? We broke down the problem into manageable steps, and by carefully following those steps, we arrived at the correct answer. Solving word problems with systems of equations might seem daunting at first, but with practice, you'll become a pro at it. Remember, the key is to read the problem carefully, identify the unknowns, translate the information into equations, solve the system, check your solution, and then clearly state your final answer. Keep practicing, and you'll master these types of problems in no time!

Key Takeaways

So, what have we learned today, guys? Let's recap the key takeaways from solving this word problem using a system of equations:

  1. Read Carefully and Identify Unknowns: The very first step is to read the problem thoroughly and pinpoint what you need to find. In our case, we needed to find the price of the book and the notebook. Identifying these unknowns is crucial because they will become your variables.
  2. Translate Words into Equations: This is where the magic happens! Convert the given information into mathematical equations. Look for relationships and quantities that you can express with symbols and numbers. Remember, each piece of information can often be translated into an equation.
  3. Set up the System of Equations: Once you have your equations, you'll have a system of equations. This is a set of two or more equations that you'll solve simultaneously. Make sure your equations accurately represent the problem's conditions.
  4. Choose a Solving Method (Substitution or Elimination): There are a couple of popular methods for solving systems of equations: substitution and elimination. Choose the method that seems most convenient for your particular problem. In our example, we used substitution because one of the equations already had a variable isolated.
  5. Solve for One Variable: Using your chosen method, solve the system for one variable. This will give you a numerical value for one of your unknowns.
  6. Substitute to Find the Other Variable(s): Once you have the value of one variable, substitute it back into one of the original equations to find the value of the other variable(s).
  7. Check Your Solution: This is a non-negotiable step! Plug your values back into the original equations or the problem's conditions to ensure they hold true. This will help you catch any errors and build confidence in your answer.
  8. State the Final Answer Clearly: Finally, state your answer in a clear and concise way, answering the question that was originally asked. Make sure your answer makes sense in the context of the problem.

By following these steps, you'll be well-equipped to tackle any word problem involving systems of equations. Practice makes perfect, so keep working at it, and you'll become a master problem-solver!

Practice Makes Perfect

Alright guys, now that we've walked through a detailed example and highlighted the key takeaways, it's time for the most important part: practice! Solving word problems with systems of equations, like any mathematical skill, gets easier and more intuitive with practice. The more problems you tackle, the better you'll become at recognizing patterns, setting up equations, and choosing the most efficient solving methods. Practice helps you build confidence and develop a deeper understanding of the underlying concepts. Think of it like learning a new language or a musical instrument – the more you practice, the more fluent and skilled you become. So, don't be afraid to dive into a variety of problems, experiment with different approaches, and learn from any mistakes you make along the way. Remember, every problem you solve is a step forward in your journey to mastering this valuable skill. So, grab your pencils, notebooks, and textbooks, and let's get practicing!

To help you on your practice journey, here are a few tips:

  • Start with simpler problems: Build your confidence by tackling easier problems first. As you get more comfortable, gradually increase the difficulty level.
  • Break down complex problems: If a problem seems overwhelming, break it down into smaller, more manageable parts. This will make the problem less intimidating and easier to solve.
  • Draw diagrams or charts: Visual aids can be incredibly helpful in understanding the relationships between different quantities. Try drawing diagrams or charts to represent the information given in the problem.
  • Work with a study group: Collaborating with friends or classmates can make practice more enjoyable and help you learn from each other's perspectives.
  • Don't give up!: Some problems might be challenging, but don't get discouraged. Persevere, and you'll eventually find a solution. Remember, every mistake is a learning opportunity.

So, there you have it! We've explored the ins and outs of solving word problems using systems of equations. We've broken down the process into clear steps, tackled an example problem, discussed key takeaways, and emphasized the importance of practice. Now, it's your turn to put your newfound knowledge into action. Happy problem-solving, guys! You've got this!