How To Solve For X: A Step-by-Step Guide

Ever stared at an equation and wondered, "What is x?" You're not alone! Solving for 'x' is a fundamental skill in mathematics, and it's often the first step in tackling more complex problems. Today, we're going to break down how to solve for 'x' using a common example: $3(2x + 4) = 5x + 20$. Don't let the numbers and letters intimidate you; we'll walk through it step-by-step, making it easy to understand and even a little fun. This guide is perfect for students just starting with algebra or anyone who needs a refresher on these essential mathematical techniques. We'll cover everything from distributing terms to isolating the variable, ensuring you feel confident in your ability to find that elusive 'x'. So, grab a pen and paper, and let's dive into the world of algebraic equations!

Understanding the Goal: Isolating 'x'

In any equation where you need to solve for x, your ultimate goal is to get 'x' all by itself on one side of the equals sign. Think of it like a balancing act. Whatever you do to one side of the equation, you must do the exact same thing to the other side to keep the equation true. Our equation, $3(2x + 4) = 5x + 20$, might look a bit tricky at first glance, especially with the parentheses. However, we can simplify it using a few key algebraic rules. The first step in simplifying this particular equation is to deal with those parentheses. This involves a concept called the distributive property. This property states that when you have a number multiplied by a term inside parentheses, you multiply that number by each term inside the parentheses. So, for $3(2x + 4)$, you'll multiply 3 by $2x$ and then multiply 3 by 4. This process removes the parentheses and makes the equation much easier to work with. After distributing, you'll have a new expression on the left side of the equals sign. This is a crucial step because it transforms the equation into a more manageable form, usually one without any grouping symbols, bringing you closer to isolating 'x'. Remember, the distributive property is a powerful tool in algebra, and mastering it will unlock many more equation-solving possibilities. It’s all about systematically breaking down the problem into smaller, more solvable parts. We're not just blindly following steps; we're using established mathematical principles to simplify and clarify the problem, making the path to finding 'x' clear and logical.

Step 1: Distribute the Terms

Let's start by applying the distributive property to the left side of our equation: $3(2x + 4) = 5x + 20$. The distributive property tells us to multiply the number outside the parentheses (which is 3) by each term inside the parentheses ($2x$ and $4$). So, we perform the following multiplications:

• $3 \times 2x = 6x$
• $3 \times 4 = 12$

Now, we can rewrite the equation with the parentheses removed. The left side becomes $6x + 12$. Our equation now looks like this:

$6x + 12 = 5x + 20$

See? We've already made the equation look much simpler. By distributing, we've eliminated the parentheses, which often represent a hurdle for beginners. This step is fundamental in algebra because it allows us to expand expressions and prepare them for further manipulation. It’s like clearing the path in a dense forest; once the underbrush is removed, you can see the trail ahead much more clearly. The result, $6x + 12 = 5x + 20$, is an equivalent equation to the original, meaning it has the same solutions. We haven't changed the value or the problem, just its appearance. This is the essence of algebraic manipulation: transforming an equation into a simpler, equivalent form without altering its fundamental truth. It's a process of simplification that brings us closer to our ultimate goal of finding the value of 'x'. This initial step is often the most critical, as errors here can propagate through the rest of the calculation. Therefore, it's vital to be accurate and deliberate when distributing.

Step 2: Combine 'x' Terms

Now that we've distributed, our equation is $6x + 12 = 5x + 20$. The next goal is to get all the terms containing 'x' onto one side of the equation and all the constant terms (numbers without variables) onto the other side. To do this, we can use addition and subtraction properties of equality. Let's choose to move the 'x' terms to the left side. We have $6x$ on the left and $5x$ on the right. To eliminate the $5x$ from the right side, we subtract $5x$ from both sides of the equation:

$(6x + 12) - 5x = (5x + 20) - 5x$

On the left side, $6x - 5x$ equals $1x$, or simply $x$. On the right side, $5x - 5x$ equals 0, leaving us with just 20.

So, our equation simplifies to:

$x + 12 = 20$

This step is about consolidation. We're bringing like terms together. By moving all the 'x' terms to one side, we are one step closer to isolating 'x'. Imagine you have two piles of apples (the 'x' terms) and two piles of oranges (the constant terms). This step is about gathering all the apples into one area and all the oranges into another. It's a crucial part of simplifying the equation, making it easier to see what we need to do next. The subtraction we performed on both sides ensures that the equality of the equation remains intact. Every operation we do must be balanced. If we only subtracted $5x$ from the right side, the equation would no longer be true. This careful balancing is what allows us to manipulate equations effectively. The result, $x + 12 = 20$, is a much simpler form, clearly showing the relationship between 'x' and the constants.

Step 3: Isolate the Variable

We're almost there! Our equation is now $x + 12 = 20$. The variable 'x' is almost isolated; it just has a '+ 12' next to it. To get 'x' completely by itself, we need to undo the addition of 12. We do this by subtracting 12 from both sides of the equation. This is again using the subtraction property of equality:

$(x + 12) - 12 = 20 - 12$

On the left side, $+ 12$ and $- 12$ cancel each other out, leaving just $x$. On the right side, $20 - 12$ equals $8$.

Therefore, we have:

$x = 8$

And there you have it! We have successfully solved for x. This final step is the culmination of all our efforts. By isolating the variable, we find its exact value. It's like reaching the end of a treasure hunt; all the clues and steps have led you to the hidden prize, which in this case, is the numerical value of 'x'. The process of isolating the variable is the core of solving most algebraic equations. It relies on the inverse operations – addition undoes subtraction, subtraction undoes addition, multiplication undoes division, and division undoes multiplication. In this case, we used subtraction to undo the addition. This methodical approach ensures accuracy and clarity. We've transformed a seemingly complex equation into a simple statement: $x$ is equal to $8$. This value, $8$, is the solution to the original equation $3(2x + 4) = 5x + 20$.

Step 4: Check Your Solution

It's always a good practice to check your answer to make sure you haven't made any mistakes along the way. We found that $x = 8$. Let's substitute this value back into the original equation: $3(2x + 4) = 5x + 20$. We need to see if both sides of the equation are equal when $x = 8$.

Left side: $3(2(8) + 4)$ $3(16 + 4)$ $3(20)$ $60$

Right side: $5(8) + 20$ $40 + 20$ $60$

Since the left side ($60$) equals the right side ($60$), our solution $x = 8$ is correct!

Checking your work is a vital part of the problem-solving process in mathematics. It reinforces your understanding and builds confidence. If the numbers on both sides didn't match, it would signal that an error occurred somewhere in the previous steps, prompting you to review your work. This verification step is not just about finding mistakes; it's about confirming the validity of your solution. It's like double-checking a recipe before serving a dish to ensure everything is perfect. In algebra, this check is often called substitution. By substituting the found value of the variable back into the original equation, we are essentially testing if our derived solution satisfies the initial conditions of the problem. This methodical approach to problem-solving, which includes checking, is what makes mathematics a powerful and reliable discipline. It provides a system for verifying truth and accuracy, ensuring that the conclusions we draw are sound.

Conclusion

Solving for 'x' in equations like $3(2x + 4) = 5x + 20$ is a fundamental skill in algebra that opens the door to understanding more complex mathematical concepts. We've walked through the process step-by-step: first, using the distributive property to simplify the equation, then combining like terms to gather the 'x' variables on one side, followed by isolating the variable to find its value, and finally, checking our answer by substituting it back into the original equation. Each step relies on basic algebraic principles like the properties of equality, ensuring that we maintain the balance of the equation throughout the process. Remember, practice makes perfect! The more equations you solve, the more comfortable and confident you'll become. Don't be afraid to tackle new problems; they are opportunities to learn and grow your mathematical abilities. If you're looking to deepen your understanding of algebra and equations, exploring resources like Khan Academy offers a wealth of free lessons, exercises, and videos that can further enhance your learning journey. They provide excellent explanations and practice problems covering a wide range of mathematical topics, from basic arithmetic to advanced calculus. Their structured approach can help solidify your grasp on algebraic concepts and provide additional practice opportunities. For more advanced algebraic techniques, you might also find the resources at Math is Fun to be incredibly helpful, offering clear explanations and interactive tools. They break down complex topics into digestible parts, making learning more accessible and engaging for everyone.