Solving The Linear Equation: 2y - 7 = 3

by Tom Lembong 40 views
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Let's break down how to solve the linear equation 2y - 7 = 3. If you're just diving into algebra or need a refresher, you've come to the right place! We'll go step by step to make sure you understand exactly how to isolate 'y' and find the solution. Trust me, it's easier than it looks!

Understanding Linear Equations

First off, what exactly is a linear equation? Simply put, it's an equation where the highest power of the variable (in our case, 'y') is 1. This means you won't see any y², y³, or anything like that. Linear equations are straight lines when graphed, hence the name 'linear.'

The general form of a linear equation is ax + b = c, where 'a,' 'b,' and 'c' are constants, and 'x' is the variable. In our equation, 2y - 7 = 3, 'a' is 2, 'b' is -7, 'c' is 3, and 'y' is our variable.

Linear equations are fundamental in mathematics and have tons of real-world applications. From calculating simple interest to determining the trajectory of a rocket, understanding how to solve them is super useful. Plus, mastering linear equations is a stepping stone to more advanced math concepts.

Why is it so important to grasp this concept? Because linear equations show up everywhere! Think about budgeting – if you know how much money you have and how much you spend each week, you can use a linear equation to figure out how long it will take to save up for something. Or consider cooking – scaling recipes up or down often involves linear relationships. Even in physics, understanding motion at a constant speed involves linear equations. So, getting comfortable with them is like unlocking a Swiss Army knife for problem-solving.

Step-by-Step Solution

Now, let’s get down to business and solve 2y - 7 = 3. Here’s how we do it, step by step:

Step 1: Isolate the Term with 'y'

Our goal is to get the term with 'y' (which is 2y) by itself on one side of the equation. To do this, we need to get rid of that -7. How? By adding 7 to both sides of the equation. Remember, whatever you do to one side, you have to do to the other to keep the equation balanced.

So, we start with:

2y - 7 = 3

Add 7 to both sides:

2y - 7 + 7 = 3 + 7

This simplifies to:

2y = 10

Step 2: Solve for 'y'

Great! Now we have 2y = 10. This means 2 times 'y' equals 10. To find 'y,' we need to undo the multiplication by dividing both sides of the equation by 2.

Starting with:

2y = 10

Divide both sides by 2:

2y / 2 = 10 / 2

This simplifies to:

y = 5

And that's it! We've found that y = 5 is the solution to the equation 2y - 7 = 3.

Step 3: Verification (Optional but Recommended)

To make sure we didn't make any mistakes, it's always a good idea to plug our solution back into the original equation. This is called verification. Let's see if y = 5 really works.

Original equation:

2y - 7 = 3

Substitute y = 5:

2(5) - 7 = 3

Simplify:

10 - 7 = 3

3 = 3

Yep, it checks out! Both sides of the equation are equal, so we know that y = 5 is indeed the correct solution.

Alternative Methods

While the step-by-step method is the most straightforward, there are other ways to think about solving linear equations. One handy trick is to use inverse operations. Think of it like unwrapping a present – you need to undo each step in reverse order.

In our case, the original equation is 2y - 7 = 3. To isolate 'y,' we first undo the subtraction of 7 by adding 7 to both sides. Then, we undo the multiplication by 2 by dividing both sides by 2. This approach can be particularly useful when dealing with more complex equations.

Another way to visualize this is by thinking of a balance scale. The equals sign (=) represents the point of balance. Whatever you add or subtract from one side, you must do the same to the other to maintain that balance. This mental image can help you keep track of the operations and avoid common mistakes.

Some people also find it helpful to rearrange the equation mentally before solving it on paper. For instance, you might think, "What number, when multiplied by 2 and then subtracted by 7, equals 3?" This can help you develop a sense of the solution before you even start writing anything down.

Real-World Applications

So, you might be wondering, "Okay, I can solve this equation, but when will I ever use this in real life?" Good question! Linear equations are everywhere once you start looking for them.

Example 1: Budgeting

Imagine you have a starting budget of $200, and you spend $15 each week. You want to know how many weeks it will take until you have only $50 left. You can model this situation with the equation:

200 - 15w = 50

Here, 'w' represents the number of weeks. Solving for 'w' will tell you how many weeks it takes to reach $50.

Example 2: Calculating Costs

Suppose you're planning a party and need to rent a venue. The venue charges a flat fee of $100 plus $10 per guest. If you have a budget of $500, how many guests can you invite?

The equation would be:

100 + 10g = 500

Where 'g' is the number of guests. Solving for 'g' will give you the maximum number of guests you can invite without exceeding your budget.

Example 3: Travel Time

Let's say you're driving to a city 150 miles away. You've already driven 30 miles. If you continue driving at 60 miles per hour, how much longer will it take to reach your destination?

The equation is:

30 + 60t = 150

Where 't' is the time in hours. Solving for 't' will tell you how much more time you need to drive.

These are just a few examples, but linear equations pop up in physics, engineering, economics, and many other fields. The ability to solve them is a fundamental skill that will serve you well in many areas of life.

Common Mistakes to Avoid

When solving linear equations, it's easy to make small mistakes that can lead to the wrong answer. Here are a few common pitfalls to watch out for:

  • Forgetting to apply operations to both sides: Remember, whatever you do to one side of the equation, you must do to the other. If you only add 7 to one side of 2y - 7 = 3, you'll throw off the balance and get an incorrect result.
  • Incorrectly combining like terms: Make sure you're only combining terms that are actually alike. For example, you can combine 3x + 2x to get 5x, but you can't combine 3x + 2 to get 5x.
  • Sign errors: Pay close attention to the signs (+ and -) in the equation. A simple sign error can completely change the solution. For instance, if you accidentally write 2y - 7 = -3 instead of 2y - 7 = 3, you'll end up with a different answer.
  • Dividing or multiplying by zero: This is a big no-no! Dividing by zero is undefined and will lead to nonsensical results. If you ever encounter a situation where you need to divide by a variable, make sure that variable cannot be zero.
  • Not checking your answer: Always, always, always check your solution by plugging it back into the original equation. This is the best way to catch any mistakes and ensure that you've found the correct answer.

By being aware of these common mistakes, you can avoid them and improve your accuracy when solving linear equations.

Practice Problems

Okay, guys, let's put your skills to the test! Here are a few practice problems for you to try. Work through them step by step, and don't forget to check your answers.

  1. 3x + 5 = 14
  2. 5a - 9 = 6
  3. 4z + 7 = 31
  4. 2b - 3 = 9
  5. 6y + 2 = 20

Answers: 1. x = 3, 2. a = 3, 3. z = 6, 4. b = 6, 5. y = 3

Solving linear equations is like learning a new language – the more you practice, the more fluent you'll become. So, don't be afraid to make mistakes and keep practicing. With a little effort, you'll be solving linear equations like a pro in no time!

Conclusion

So, to wrap it up, the solution to the equation 2y - 7 = 3 is y = 5. Remember, solving linear equations involves isolating the variable by using inverse operations and keeping the equation balanced. With practice and a good understanding of the basic principles, you'll be able to tackle any linear equation that comes your way. Keep practicing, and you'll become a pro in no time! You got this!