Solve Algebraic Equations: (3x+1)/10 + (2x+5)/9 = X+4

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the awesome world of mathematics, specifically tackling a rather juicy algebraic equation: $\frac{3 x+1}{10}+\frac{2 x+5}{9}=x+4$. Now, I know what some of you might be thinking – "Ugh, fractions and variables, my worst nightmare!" But don't you worry your pretty little heads about it. We're going to break this down step-by-step, making it as clear as day. By the end of this, you'll be feeling like a total math whiz, ready to conquer any equation that comes your way. So, grab your favorite beverage, get comfy, and let's get this math party started! We'll explore the nitty-gritty of isolating 'x', dealing with those pesky denominators, and ultimately finding that elusive solution. It's all about understanding the process, and trust me, it's more fun than you think!

Understanding the Equation

Alright, let's first get a solid grasp on the equation we're working with: $\frac{3 x+1}{10}+\frac{2 x+5}{9}=x+4$. What we're trying to do here, in essence, is to find the value of 'x' that makes this entire statement true. Think of it like a balancing scale; whatever we do to one side, we must do to the other to keep it perfectly balanced. The trickiest part for many of you, I bet, is dealing with those fractions. We've got a denominator of 10 on the left side and a denominator of 9 on the left side as well. On the right side, we have a simple 'x' and a '4', which can be thought of as $\frac{x+4}{1}$. Our main goal is to simplify this equation until we have 'x' all by itself on one side. This means we need to get rid of those denominators. We’ll explore different strategies for this, but the most common and effective way is to find a common denominator. This is a number that both 10 and 9 can divide into evenly. Once we have that, we can multiply the entire equation by this common denominator to clear out the fractions. It sounds a bit daunting, but it's a super powerful technique that simplifies things dramatically. We'll also be looking at combining like terms, which is another fundamental skill in algebra. This means grouping all the terms that have 'x' in them together and grouping all the constant numbers together. This systematic approach will lead us smoothly towards our solution. Remember, patience is key! Math is like building with LEGOs; you have to place each brick carefully, and before you know it, you have something amazing. So, let's focus on each step, and we'll nail this equation.

Finding the Common Denominator

So, how do we banish those pesky fractions? The secret sauce, my friends, is finding a common denominator. For our equation $\frac{3 x+1}{10}+\frac{2 x+5}{9}=x+4$, we have denominators of 10 and 9. We need to find the smallest number that both 10 and 9 can divide into evenly. This is called the Least Common Multiple (LCM). To find the LCM of 10 and 9, we can list out their multiples:

Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, ... Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, ...

See that? The smallest number that appears in both lists is 90! So, 90 is our Least Common Multiple. This is fantastic news because it means we can now multiply every single term in our equation by 90. Why does this work, you ask? Because multiplying by 90 is like multiplying by $\frac{90}{1}$. When we multiply a fraction by a number that is a multiple of its denominator, the denominator cancels out. For instance, when we multiply $\frac{3x+1}{10}$ by 90, we can think of it as $\frac{90}{1} \times \frac{3x+1}{10}$. The 90 and the 10 share a common factor of 10. $\frac{90}{10}$ simplifies to 9. So, $\frac{3x+1}{10} \times 90$ becomes $9 \times (3x+1)$. We'll do the same for the other fraction. This process is absolutely key to simplifying our equation and making it much easier to solve. It's like performing a magic trick where the fractions just disappear, leaving us with a cleaner, more manageable expression. Remember, whatever you do to one side of the equation, you must do to the other. So, we'll multiply the right side ($x+4$) by 90 as well. This consistency ensures our equation remains balanced and our solution accurate. This step is crucial, so pay close attention to how the denominators vanish!

Clearing the Fractions

Now that we've identified our common denominator as 90, it's time for the magic to happen – clearing those fractions! We're going to multiply each term in our equation $\frac{3 x+1}{10}+\frac{2 x+5}{9}=x+4$ by 90. Let's do this carefully, term by term:

1. Multiply the first term by 90: $90 \times \frac{3 x+1}{10} = \frac{90}{10} \times (3x+1) = 9 \times (3x+1)$

2. Multiply the second term by 90: $90 \times \frac{2 x+5}{9} = \frac{90}{9} \times (2x+5) = 10 \times (2x+5)$

3. Multiply the term 'x' on the right side by 90: $90 \times x = 90x$

4. Multiply the constant '4' on the right side by 90: $90 \times 4 = 360$

So, after multiplying the entire equation by 90, we get a new, fraction-free equation:

$9 \times (3x+1) + 10 \times (2x+5) = 90x + 360$

See how the denominators are gone? Poof! This is where things get much simpler. The next step is to distribute the numbers outside the parentheses. We'll multiply the 9 by each term inside the first parenthesis and the 10 by each term inside the second parenthesis. This is a fundamental algebraic step that allows us to expand and simplify expressions. Remember the distributive property: $a(b+c) = ab + ac$. We apply this property to both parts of our equation. This process is crucial for preparing the equation for the next stage, where we'll combine like terms. Take your time with the distribution; it's easy to make small errors here, but with a little focus, you'll get it right. This step transforms our equation into a linear form, which is much easier to solve. It’s all about simplifying the complexity step-by-step, making the problem progressively more approachable. This is where the