Solve: Two Numbers Sum To 74, One Is 10 More

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Let's dive into solving this classic math problem using linear equations. This type of problem is super common and a great way to practice your algebra skills. We'll break it down step-by-step, so it's easy to follow. Guys, don't worry if you find these a bit tricky at first; with practice, you'll become a pro in no time!

Setting Up the Equations

Okay, so the heart of solving any word problem lies in translating the words into mathematical equations. In this case, we're given two key pieces of information. First, the sum of two numbers is 74. Second, one number is 10 more than the other. Let's translate these into equations.

Let's call our two numbers 'x' and 'y'. It doesn't matter which one is which for now.

  • Equation 1 (The sum): x + y = 74
  • Equation 2 (One is 10 more than the other): Let's say x = y + 10 (This means 'x' is 10 more than 'y')

Now we have a system of two linear equations with two variables. This is something we can definitely solve!

Solving the System of Equations

There are a couple of ways to solve a system of equations like this: substitution and elimination. For this problem, substitution is probably the easiest method. Since we already have x expressed in terms of y in Equation 2 (x = y + 10), we can substitute this expression for x in Equation 1.

  • Substitute: Replace 'x' in Equation 1 with 'y + 10' (y + 10) + y = 74

Now we have a single equation with just one variable, 'y'. Let's simplify and solve for 'y'.

  • Simplify: Combine the 'y' terms. 2y + 10 = 74
  • Isolate the 'y' term: Subtract 10 from both sides of the equation. 2y = 64
  • Solve for 'y': Divide both sides by 2. y = 32

Great! We've found the value of 'y'. Now that we know 'y', we can easily find 'x' using either Equation 1 or Equation 2. Let's use Equation 2 since it's already set up to solve for 'x'.

  • Substitute 'y' into Equation 2: x = 32 + 10
  • Solve for 'x': x = 42

So, we've found that x = 42 and y = 32.

Checking Our Solution

It's always a good idea to check our solution to make sure it's correct. Let's plug our values for 'x' and 'y' back into our original equations.

  • Equation 1 (x + y = 74): 42 + 32 = 74 (This is correct!)
  • Equation 2 (x = y + 10): 42 = 32 + 10 (This is also correct!)

Since our values satisfy both equations, we know our solution is correct.

The Answer

The two numbers are 42 and 32. That wasn't so bad, right? Remember, the key is to carefully translate the word problem into mathematical equations and then use your algebra skills to solve for the unknowns. Keep practicing, and you'll become a master at these types of problems!

Why This Works: A Deeper Dive

Understanding why this method works is just as important as knowing how to do it. Here’s a breakdown of the logic behind using linear equations to solve this problem.

Translating Words to Math

The most crucial step is accurately converting the problem's wording into mathematical statements. This involves identifying the unknowns (the numbers we need to find) and the relationships between them.

  • Identifying Unknowns: We assign variables (like 'x' and 'y') to represent the unknown numbers. This allows us to manipulate them algebraically.
  • Expressing Relationships: The problem provides relationships between the unknowns. For example, "the sum of two numbers is 74" translates directly to the equation x + y = 74. Similarly, "one number is 10 more than the other" becomes x = y + 10 (or y = x + 10, depending on how you define 'x' and 'y').

The Power of Systems of Equations

A system of equations allows us to solve for multiple unknowns simultaneously. To solve for 'n' unknowns, you generally need 'n' independent equations. In our case, we have two unknowns ('x' and 'y') and two equations. This gives us a solvable system.

The Substitution Method

The substitution method is a powerful technique for solving systems of equations. It involves the following steps:

  1. Isolate a Variable: Choose one equation and solve for one variable in terms of the other. In our example, Equation 2 (x = y + 10) already had 'x' isolated.
  2. Substitute: Substitute the expression you found in step 1 into the other equation. This eliminates one variable, leaving you with a single equation in one variable.
  3. Solve: Solve the resulting equation for the remaining variable.
  4. Back-Substitute: Substitute the value you found in step 3 back into one of the original equations (or the expression from step 1) to solve for the other variable.

Why Substitution Works

The substitution method works because it leverages the relationships defined by the equations. By substituting one expression for another, we're essentially combining the information from both equations into a single, solvable statement. This allows us to find the values of the unknowns that satisfy all the given conditions.

Checking the Solution: Ensuring Accuracy

Checking your solution is not just a formality; it's a critical step in the problem-solving process. By plugging the values you found back into the original equations, you verify that your solution is consistent with all the given information. If the values don't satisfy the equations, it indicates an error in your calculations or setup.

Real-World Applications

While this specific problem might seem purely academic, the underlying principles of solving systems of equations have wide-ranging applications in various fields.

  • Engineering: Engineers use systems of equations to analyze circuits, design structures, and model complex systems.
  • Economics: Economists use systems of equations to model supply and demand, analyze market equilibrium, and forecast economic trends.
  • Computer Science: Computer scientists use systems of equations in areas like computer graphics, optimization, and machine learning.
  • Finance: Financial analysts use systems of equations to model investments, manage risk, and value assets.
  • Everyday Life: Even in everyday situations, we often implicitly use systems of equations to make decisions, such as budgeting, planning trips, and comparing prices.

For example, imagine you're planning a road trip and need to decide how much to spend on gas versus lodging. You have a total budget, and you know the approximate cost per gallon of gas and the average cost of a hotel room. You can set up a system of equations to determine the optimal allocation of your budget to maximize your trip's enjoyment.

Tips and Tricks for Solving Linear Equations

Here are some handy tips and tricks to make solving linear equations easier and more efficient:

  • Read Carefully: Always read the problem statement carefully and identify the key information and unknowns.
  • Define Variables Clearly: Clearly define your variables and what they represent. This will help you avoid confusion and make the equations easier to understand.
  • Write Equations Systematically: Write down your equations in a clear and organized manner. This will make it easier to spot errors and keep track of your work.
  • Choose the Right Method: Select the most appropriate method for solving the system of equations. Substitution is often a good choice when one variable is already isolated or can be easily isolated. Elimination is useful when the coefficients of one variable are the same or opposites in the two equations.
  • Simplify Before Solving: Simplify the equations as much as possible before attempting to solve them. This can make the calculations easier and reduce the chance of errors.
  • Check Your Work Regularly: Check your work at each step to ensure that you haven't made any mistakes. This will help you catch errors early on and avoid wasting time on incorrect calculations.
  • Practice Regularly: The best way to improve your skills in solving linear equations is to practice regularly. Work through a variety of problems and try different approaches. The more you practice, the more comfortable and confident you'll become.

Conclusion

So, there you have it! Solving the problem of finding two numbers that sum to 74, where one is 10 more than the other, is a perfect example of how linear equations can be used to solve real-world problems. By translating the word problem into mathematical equations, using the substitution method, and checking our solution, we were able to find the answer with confidence. Remember to practice regularly and apply these techniques to other problems to further develop your algebra skills. Keep up the great work, and you'll become a math whiz in no time! Remember guys, the more you practice the better you get.