Solving For Y: A = 9y + 3yx Explained Simply

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Hey guys! Today, we're diving into a common algebra problem: solving a literal equation for a specific variable. In this case, we want to isolate y in the equation a = 9y + 3yx. Don't worry, it's not as intimidating as it looks! We'll break it down step-by-step, so you can tackle similar problems with confidence. So buckle up and let's get started!

Understanding Literal Equations

Before we jump into the solution, let's quickly define what a literal equation is. A literal equation is simply an equation where the coefficients and constants are represented by letters. These letters, other than the variable we're solving for, are often called parameters. The goal is to isolate the desired variable on one side of the equation, expressing it in terms of the other variables.

Literal equations are super useful because they allow us to create general formulas. For instance, the equation a = 9y + 3yx might represent a relationship between area (a), a dimension y, and another variable x in a geometric context. Solving for y allows us to find the value of that dimension if we know the area and the value of x. Understanding literal equations and how to manipulate them is a fundamental skill in algebra and beyond.

Why are literal equations important? Well, they pop up everywhere! Physics, engineering, economics – you name it. Being able to rearrange formulas quickly saves you time and effort. Instead of plugging in numbers and then solving, you can rearrange the formula once and then plug in different values as needed. This is especially helpful when dealing with complex formulas or repeated calculations. Think of it as creating a customized tool for your specific problem. Plus, it demonstrates a strong understanding of algebraic principles, which is always a good thing!

Step-by-Step Solution

Now, let's get our hands dirty and solve the equation a = 9y + 3yx for y. Here's a breakdown of the steps:

1. Identify the terms containing y

In our equation, the terms 9y and 3yx both contain the variable y.

2. Factor out y

This is a crucial step! We factor out y from both terms on the right side of the equation:
a = y(9 + 3x)

See how we've essentially reversed the distributive property? By factoring out y, we've grouped all the terms containing y together.

3. Isolate y by dividing

To isolate y, we need to get rid of the (9 + 3x) term that's multiplying it. We do this by dividing both sides of the equation by (9 + 3x):

a / (9 + 3x) = y(9 + 3x) / (9 + 3x)

This simplifies to:

y = a / (9 + 3x)

And that's it! We've successfully solved for y.

4. The Final Answer

Therefore, the solution to the equation a = 9y + 3yx for y is:

y = a / (9 + 3x)

Example Time!

Let's say a = 36 and x = 1. What is the value of y? We can use the formula we just derived:

y = a / (9 + 3x)

Substitute the values of a and x:

y = 36 / (9 + 3 * 1)

Simplify:

y = 36 / (9 + 3)

y = 36 / 12

y = 3

So, when a = 36 and x = 1, the value of y is 3.

Common Mistakes to Avoid

When solving literal equations, it's easy to make mistakes. Here are a few common pitfalls to watch out for:

  • Forgetting to factor: Make sure you factor out the variable you're solving for correctly. Double-check that when you distribute the variable back into the parentheses, you get the original expression.
  • Incorrectly dividing: Ensure you divide the entire side of the equation by the term you're using to isolate the variable. Don't just divide individual terms.
  • Combining unlike terms: You can only combine terms that have the same variables and exponents. For example, you can't combine 9 and 3x directly.
  • Not simplifying: Always simplify your answer as much as possible. This makes it easier to work with and understand.

By being mindful of these common mistakes, you can increase your accuracy and avoid unnecessary errors.

Practice Problems

Want to test your understanding? Try solving these literal equations for the indicated variable:

  1. Solve for x: z = 4x + 2xw
  2. Solve for p: r = 5p - 3pq
  3. Solve for m: n = 2m + 7mk

Solving these practice problems will solidify your understanding of the steps involved and help you develop your problem-solving skills. Don't be afraid to make mistakes – that's how you learn! And if you get stuck, review the steps we discussed earlier or seek help from a tutor or online resource.

Conclusion

Solving literal equations for a specific variable, like y in the equation a = 9y + 3yx, is a fundamental skill in algebra. By understanding the steps involved – identifying terms containing the variable, factoring, and isolating the variable – you can confidently tackle these types of problems. Remember to avoid common mistakes, practice regularly, and don't be afraid to ask for help when needed. Keep practicing, and you'll become a pro at solving literal equations in no time! You got this! Now you have the knowledge to solve literal equations and even teach them to your friends.