Solving For X: 3/(x+3) = 9/(2x+11) | Step-by-Step Guide
Hey everyone! Today, we're diving into a fun math problem where we need to solve for x in the equation 3/(x+3) = 9/(2x+11). This type of problem involves fractions and variables, but don't worry, we'll break it down step-by-step so it's super easy to understand. So, grab your pencils, and let's get started!
Understanding the Problem
Before we jump into solving, let's quickly understand what the equation is asking us. We have two fractions that are equal to each other. Our mission is to find the value of x that makes this equation true. In other words, we want to find the number that, when we substitute it for x, will make both sides of the equation equal. This is a fundamental concept in algebra, and mastering it will help you tackle more complex problems later on.
The core of solving this equation lies in understanding how to manipulate it while maintaining the equality. We'll use techniques like cross-multiplication and simplifying to isolate x on one side of the equation. Remember, the key is to perform the same operations on both sides to keep the equation balanced. This ensures that the value of x we find is the correct solution. So, let's dive into the step-by-step process of cracking this equation!
Setting up the Solution
To kick things off, let's rewrite the equation so we have it clearly in front of us: 3/(x+3) = 9/(2x+11). The first thing we want to do is get rid of the fractions. The best way to do this is by using a technique called cross-multiplication. Cross-multiplication is a handy shortcut when you have two fractions equal to each other. It helps us eliminate the denominators and turn the equation into a more manageable form. By applying this method, we are essentially multiplying both sides of the equation by the denominators, but in a streamlined way.
Cross-multiplication makes solving equations with fractions way easier. It transforms our equation into a linear equation, which is something we can solve using basic algebra. Think of it like building a bridge across the equals sign – you're connecting the numerators of one fraction with the denominators of the other. This simple trick is a cornerstone of algebraic problem-solving, and you'll find it incredibly useful in many different scenarios. So, let's see how cross-multiplication works in action with our specific equation.
Step-by-Step Solution
Now, let's get down to the nitty-gritty and solve this equation! We'll take it one step at a time, so you can follow along easily.
1. Cross-Multiplication
The first step, as we mentioned, is cross-multiplication. This means we'll multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa. So, we'll multiply 3 by (2x + 11) and 9 by (x + 3). This gives us:
3 * (2x + 11) = 9 * (x + 3)
Cross-multiplication is like a magic wand for fraction equations, instantly turning them into something less scary. What we've essentially done is multiply both sides of the equation by (x + 3) and by (2x + 11). This eliminates the fractions, leaving us with a cleaner equation to work with. This technique is not just a trick; it's based on the fundamental principles of algebraic manipulation. By understanding why it works, you're building a solid foundation for tackling more complex equations in the future.
2. Distribute
Next, we need to get rid of the parentheses. To do this, we'll use the distributive property, which means we'll multiply the numbers outside the parentheses by each term inside the parentheses.
For the left side of the equation, we have 3 * (2x + 11). So, we'll multiply 3 by 2x and 3 by 11:
3 * 2x = 6x 3 * 11 = 33
So, the left side becomes 6x + 33.
Now, let's do the same for the right side of the equation, where we have 9 * (x + 3). We'll multiply 9 by x and 9 by 3:
9 * x = 9x 9 * 3 = 27
The right side becomes 9x + 27.
Now, our equation looks like this:
6x + 33 = 9x + 27
Distribution is a crucial step in simplifying algebraic expressions. It allows us to break down complex terms into smaller, more manageable pieces. By correctly applying the distributive property, we ensure that each term within the parentheses is properly accounted for. This step sets the stage for isolating the variable, which is our ultimate goal. Remember, the distributive property is your friend when you see parentheses in equations – it's the key to unlocking the next step in the solution.
3. Isolate the Variable
Our next goal is to get all the x terms on one side of the equation and all the constant terms (the numbers without x) on the other side. This is called isolating the variable.
Let's start by moving the x terms. We have 6x on the left side and 9x on the right side. To get the x terms on one side, we can subtract 6x from both sides of the equation. This will eliminate the 6x term on the left side:
6x + 33 - 6x = 9x + 27 - 6x
This simplifies to:
33 = 3x + 27
Now, let's move the constant terms. We have 33 on the left side and 27 on the right side. To get the constant terms on one side, we can subtract 27 from both sides of the equation. This will eliminate the 27 on the right side:
33 - 27 = 3x + 27 - 27
This simplifies to:
6 = 3x
Isolating the variable is a fundamental technique in algebra. It's like solving a puzzle where you're trying to get one specific piece (the variable) by itself. By performing the same operations on both sides of the equation, we maintain the balance and ensure that we're moving closer to the correct solution. This step is where we start to see the value of x emerge, and it's a crucial step in solving for the unknown.
4. Solve for x
We're almost there! We now have the equation 6 = 3x. To solve for x, we need to get x by itself. Right now, x is being multiplied by 3. To undo this multiplication, we'll divide both sides of the equation by 3:
6 / 3 = (3x) / 3
This simplifies to:
2 = x
So, we've found that x equals 2!
Solving for x is the grand finale of our algebraic journey. It's the moment where all our hard work pays off and we uncover the value of the unknown. By performing the inverse operation (in this case, division) we isolate x and reveal its true value. This step is the culmination of all the previous steps, and it's a testament to the power of algebraic manipulation. With x now in our grasp, we've successfully conquered the equation!
Checking Our Answer
It's always a good idea to check our answer to make sure it's correct. To do this, we'll substitute our value for x (which is 2) back into the original equation and see if both sides are equal.
Our original equation was:
3/(x+3) = 9/(2x+11)
Now, let's substitute x with 2:
3/(2+3) = 9/(2*2+11)
Simplify the expressions inside the parentheses:
3/5 = 9/(4+11)
Simplify further:
3/5 = 9/15
Now, we can simplify the right side of the equation by dividing both the numerator and the denominator by 3:
9/15 = (9/3) / (15/3) = 3/5
So, we have:
3/5 = 3/5
Since both sides of the equation are equal, our answer is correct!
Checking our answer is like putting the final piece of a puzzle in place – it confirms that everything fits together perfectly. By substituting our solution back into the original equation, we ensure that we haven't made any mistakes along the way. This step is not just about getting the right answer; it's about building confidence in our problem-solving skills. It's a testament to the fact that we've not only solved the equation but also verified its correctness. So, always remember to check your work – it's the hallmark of a true math whiz!
Conclusion
Awesome! We've successfully solved for x in the equation 3/(x+3) = 9/(2x+11), and we found that x = 2. We did this by using cross-multiplication, distributing, isolating the variable, and then solving for x. And, we didn't forget to check our answer to make sure we were right!
Solving equations like this might seem tricky at first, but with practice, you'll become a pro! Remember the key steps: clear the fractions, simplify, isolate the variable, and solve. And always double-check your work – it's like having a superpower that ensures your success. Keep practicing, and you'll be tackling even tougher math challenges in no time. You've got this!
If you have any questions or want to try another problem, just let me know. Happy solving!