Easy Math: Solve 2x - 3 + 4x = 21

Hey guys! Let's dive into a super common type of math problem you'll see – solving linear equations. Today, we've got this gem: $2x - 3 + 4x = 21$. Don't let the letters and numbers get you down; we'll break it down step-by-step so it's as clear as mud (just kidding, it'll be crystal clear!). These kinds of problems are fundamental in algebra, and once you get the hang of them, you'll feel like a math whiz. We're going to simplify the equation first, then isolate the variable '$x$' to find its value. Remember, the goal in solving an equation is to get the variable all by itself on one side of the equals sign. Think of the equals sign as a balancing scale; whatever you do to one side, you must do to the other to keep it balanced. This principle is key to solving pretty much any equation you'll encounter. So, grab your favorite thinking cap, maybe a snack, and let's get this done!

Step 1: Combine Like Terms

The very first thing we want to do when we look at the equation $2x - 3 + 4x = 21$ is to simplify it as much as possible. Notice that we have two terms with '$x$' in them: '$2x$' and '$4x$'. These are what we call 'like terms' because they both contain the same variable raised to the same power (in this case, $x^1$). We can combine these like terms by simply adding their coefficients (the numbers in front of the '$x$'). So, $2x + 4x$ becomes $6x$. Now, our equation looks a little cleaner: $6x - 3 = 21$. It's already looking less intimidating, right? Combining like terms is a crucial first step because it reduces the number of terms you have to work with, making the subsequent steps much smoother. It's like tidying up your workspace before starting a big project; the more organized you are, the easier the task becomes. Always scan your equation for terms that can be combined – whether they are '$x$' terms, constant numbers, or even other variables. This algebraic housekeeping saves time and reduces the chance of silly mistakes later on. So, in our case, $2x$ and $4x$ are buddies, and they decided to team up to become $6x$. The constant term, '-3', stays as it is for now. The right side of the equation, '21', also remains unchanged as it's just a number on its own. The simplified equation is now $6x - 3 = 21$. Keep this simplified form in mind, as it's the foundation for our next move.

Step 2: Isolate the Variable Term

Okay, we've simplified our equation to $6x - 3 = 21$. Our next mission, should we choose to accept it, is to get the term containing '$x$' (which is $6x$) all by itself on one side. Right now, it's being 'hindered' by the '-3'. To get rid of that '-3', we need to perform the opposite operation. Since it's currently being subtracted, we'll do the opposite: add 3. And remember our balancing scale rule? Whatever we do to one side, we must do to the other. So, we'll add 3 to both sides of the equation:

$6x - 3 + 3 = 21 + 3$

On the left side, the '-3' and '+3' cancel each other out, leaving us with just $6x$. On the right side, $21 + 3$ equals $24$. So, our equation is now dramatically simpler:

$6x = 24$

See how much progress we've made? We've successfully isolated the term that has our variable '$x$' in it. This step is all about undoing any addition or subtraction that's attached to the variable term. If you had something like $6x + 5 = 20$, you'd subtract 5 from both sides. If you had $6x + 10 = 15$, you'd subtract 10. The key is to identify the constant number that's either added or subtracted from your variable term and perform the inverse operation on both sides of the equation. This keeps the equality true and moves us closer to finding the value of '$x$'. In our specific problem, the '-3' was the obstacle, and adding 3 to both sides was the solution to clearing it out of the way, leaving us with $6x = 24$.

Step 3: Solve for x

Alright, we're in the home stretch, guys! We've got our equation down to $6x = 24$. This means '6 multiplied by $x$' equals 24. To find out what '$x$' is, we need to undo the multiplication. The opposite of multiplying by 6 is dividing by 6. So, we'll divide both sides of the equation by 6:

$$\frac{6x}{6} = \frac{24}{6}$$

On the left side, the '6' in the numerator and the '6' in the denominator cancel each other out, leaving us with just '$x$'. On the right side, $24$ divided by $6$ is $4$. So, we get:

$x = 4$

And there you have it! We've successfully solved the equation. The value of '$x$' that makes the original equation $2x - 3 + 4x = 21$ true is $4$. This final step always involves undoing multiplication or division. If you had something like $3x = 15$, you'd divide both sides by 3. If you had $x/2 = 5$, you'd multiply both sides by 2. Always perform the inverse operation to isolate the variable. To be absolutely sure, you can always plug your answer back into the original equation. Let's do that:

Original equation: $2x - 3 + 4x = 21$

Substitute $x = 4$: $2(4) - 3 + 4(4) = 21$

Calculate: $8 - 3 + 16 = 21$

Simplify: $5 + 16 = 21$

Check: $21 = 21$

It checks out! Our solution is correct. So, the answer to the equation $2x - 3 + 4x = 21$ is $x=4$. This corresponds to option C. Awesome job, mathletes!

Conclusion

Solving linear equations like $2x - 3 + 4x = 21$ is a fundamental skill in mathematics, and as you saw, it's a straightforward process when you break it down. We started by combining like terms ($2x + 4x = 6x$) to simplify the equation to $6x - 3 = 21$. Then, we isolated the variable term by adding 3 to both sides, resulting in $6x = 24$. Finally, we solved for '$x$' by dividing both sides by 6, which gave us our answer: $x=4$. Remember the core principles: simplify, isolate the variable term, and solve for the variable. Always perform the same operation on both sides to maintain equality. And don't forget the power of checking your work by substituting your answer back into the original equation – it's your foolproof way to ensure accuracy. Keep practicing these types of problems, and you'll master them in no time. Math is all about practice and building that confidence. Keep up the great work, everyone!