Solve 3/4x = 9 For X = 12

Oct 29, 2025 by ADMIN 26 views

Hey guys! Ever stared at an equation like $rac{3}{4} x = 9$ and wondered how on earth you're supposed to get to the answer $x = 12$ ? It seems like a bit of a leap, right? Well, buckle up, because we're about to break down the magic behind transforming this equation into its simpler form. The key, my friends, lies in understanding how to manipulate equations using multiplication. We're not just randomly picking numbers here; there's a specific reason why we multiply both sides by a particular fraction. Our main goal is to isolate 'x', to get it all by its lonesome on one side of the equals sign. Think of it like trying to get a prized possession out of a locked box – you need the right key. In this case, the 'key' is the reciprocal of the coefficient (the number in front) of 'x'. The coefficient of 'x' in our equation is $rac{3}{4}$ . To get rid of it, we need to multiply by its reciprocal, which is $rac{4}{3}$ . Why? Because when you multiply a fraction by its reciprocal, the result is always 1 ( $rac{3}{4} imes rac{4}{3} = rac{12}{12} = 1$ ). So, when we multiply $rac{3}{4}x$ by $rac{4}{3}$ , we're left with $1x$ , or just $x$ . It's like canceling out the fraction entirely! And remember the golden rule of algebra: whatever you do to one side of the equation, you absolutely MUST do to the other side. This keeps the equation balanced, fair, and true. So, we'll multiply both sides of $rac{3}{4}x = 9$ by $rac{4}{3}$ . On the left side, $rac{4}{3} imes rac{3}{4}x$ becomes $1x$ , or $x$ . On the right side, we have $9 imes rac{4}{3}$ . To multiply a whole number by a fraction, you can think of the whole number as a fraction with a denominator of 1 (so, $9 = rac{9}{1}$ ). Then, you multiply the numerators together and the denominators together: $rac{9}{1} imes rac{4}{3} = rac{9 imes 4}{1 imes 3} = rac{36}{3}$ . Now, we simplify that fraction: $36 ext{ divided by } 3$ is indeed $12$ . Voilà! We've transformed $rac{3}{4}x = 9$ into $x = 12$ . The number we multiplied by, the crucial player in this transformation, was $rac{4}{3}$ . This reciprocal trick is a fundamental concept in solving linear equations, and once you get the hang of it, you'll be zipping through algebra problems like a pro. So, next time you see a fraction attached to your variable, just remember to grab its reciprocal and multiply both sides – easy peasy!

Understanding the Core Concept: Isolating the Variable

Alright, let's dive a little deeper into why this whole process works, because understanding the 'why' is super important, guys. Our main mission, should we choose to accept it (and we always should in math!), is to isolate the variable, which in this case is 'x'. Imagine 'x' is a celebrity and the equation is a crowd trying to get to them. Our job is to clear away all the fans (the $rac{3}{4}$ ) so 'x' can stand alone. The equation $rac{3}{4}x = 9$ tells us that three-quarters of 'x' is equal to 9. We want to know what the whole 'x' is. Think about it this way: if $rac{3}{4}$ of something is 9, then one whole of that thing must be bigger than 9, right? This is where the concept of multiplicative inverses, or reciprocals, comes into play. A reciprocal is simply a fraction flipped upside down. The reciprocal of $rac{a}{b}$ is $rac{b}{a}$ . When you multiply a number by its reciprocal, the result is always 1. For example, $rac{3}{4} imes rac{4}{3} = rac{12}{12} = 1$ . This property is incredibly powerful because multiplying any number by 1 doesn't change its value (that's the identity property of multiplication). So, when we multiply $rac{3}{4}x$ by $rac{4}{3}$ , we get: $rac{4}{3} imes rac{3}{4}x = ( rac{4}{3} imes rac{3}{4})x = 1x = x$ . We've successfully isolated 'x'! Now, here's the crucial part: the balance principle of equations. An equation is like a perfectly balanced scale. If you add weight to one side, you must add the exact same weight to the other side to keep it balanced. If you remove weight from one side, you must remove the same amount from the other. In our case, we multiplied the left side by $rac{4}{3}$ . To maintain the balance, we must multiply the right side by the exact same number, $rac{4}{3}$ . So, the right side becomes $9 imes rac{4}{3}$ . We can write 9 as $rac{9}{1}$ to make the multiplication clearer: $rac{9}{1} imes rac{4}{3} = rac{9 imes 4}{1 imes 3} = rac{36}{3}$ . Simplifying $rac{36}{3}$ gives us 12. So, we have $x = 12$ . The number we needed to multiply both sides by to achieve this was $rac{4}{3}$ . It's the reciprocal of $rac{3}{4}$ , and its purpose is to 'undo' the multiplication of $rac{3}{4}$ on 'x', leaving 'x' by itself. Pretty neat, huh? This method is a fundamental building block for tackling more complex algebraic problems down the line.

Why Other Options Don't Work

Now, let's quickly chat about why the other choices offered – $- rac{3}{4}$ , $rac{1}{4}$ , and $4$ – wouldn't get us to our desired $x = 12$ result when multiplying $rac{3}{4}x = 9$ . Understanding why something doesn't work is just as valuable as knowing why it does work, right? It helps solidify your grasp on the underlying principles. First up, let's look at option A: $- rac{3}{4}$ . If we multiply both sides of $rac{3}{4}x = 9$ by $- rac{3}{4}$ , we get: Left side: $(- rac{3}{4}) imes ( rac{3}{4}x) = (- rac{3}{4} imes rac{3}{4})x = - rac{9}{16}x$ . Right side: $9 imes (- rac{3}{4}) = - rac{27}{4}$ . So, we'd end up with $- rac{9}{16}x = - rac{27}{4}$ . This isn't $x = 12$ ; in fact, we've just made things more complicated by introducing a negative sign and a different fraction. We haven't isolated 'x', and we certainly haven't reached our target. The negative sign here is the biggest giveaway that it's not the right path, as our original coefficient for 'x' was positive. Next, consider option B: $rac{1}{4}$ . Multiplying both sides of $rac{3}{4}x = 9$ by $rac{1}{4}$ gives us: Left side: $rac{1}{4} imes rac{3}{4}x = rac{3}{16}x$ . Right side: $9 imes rac{1}{4} = rac{9}{4}$ . This results in $rac{3}{16}x = rac{9}{4}$ . Again, 'x' is still not isolated, and the resulting equation is far from $x = 12$ . We've essentially made the coefficient of 'x' even smaller, moving us further away from isolating it. The goal is to make the coefficient of 'x' equal to 1, and multiplying by $rac{1}{4}$ certainly doesn't achieve that when we started with $rac{3}{4}$ . Finally, let's examine option D: $4$ . If we multiply both sides by 4: Left side: $4 imes rac{3}{4}x = ( rac{4}{1} imes rac{3}{4})x = rac{12}{4}x = 3x$ . Right side: $9 imes 4 = 36$ . This gives us $3x = 36$ . Now, we're closer because we don't have a fraction anymore, but 'x' is still not isolated. We would then need another step to divide both sides by 3 to get $x = 12$ . So, while multiplying by 4 gets us part of the way there, it's not the single number that, when multiplied, directly produces the equivalent equation $x = 12$ . The number that does this in one step is the reciprocal, $rac{4}{3}$ , which effectively cancels out the $rac{3}{4}$ coefficient. So, the only option that correctly transforms $rac{3}{4}x = 9$ directly into $x = 12$ in one multiplication step is $rac{4}{3}$ . It's all about finding that perfect reciprocal to achieve isolation in a single move! Keep practicing, and you'll master this in no time, guys!

Step-by-Step Solution

Let's recap and lay out the definitive steps to solve $rac{3}{4}x = 9$ for $x=12$ . The question asks which number we should multiply each side of the equation by to get the equivalent equation $x = 12$ . The key is to eliminate the coefficient of 'x', which is $rac{3}{4}$ .

Identify the coefficient of x: In the equation $rac{3}{4}x = 9$ , the coefficient of 'x' is $rac{3}{4}$ .
Determine the reciprocal: To isolate 'x', we need to multiply by its reciprocal. The reciprocal of $rac{3}{4}$ is $rac{4}{3}$ .
Multiply both sides by the reciprocal: We must perform the same operation on both sides of the equation to maintain equality.
- Left side: $rac{4}{3} imes rac{3}{4}x$
- Right side: $9 imes rac{4}{3}$
Simplify the left side: Multiplying the coefficient by its reciprocal results in 1.
- $rac{4}{3} imes rac{3}{4}x = ( rac{4 imes 3}{3 imes 4})x = rac{12}{12}x = 1x = x$
Simplify the right side: Multiply the whole number by the fraction.
- $9 imes rac{4}{3} = rac{9}{1} imes rac{4}{3} = rac{9 imes 4}{1 imes 3} = rac{36}{3}$
Further simplify the right side: Perform the division.
- $rac{36}{3} = 12$
Combine the results: Setting the simplified left side equal to the simplified right side gives us the equivalent equation:
- $x = 12$

Therefore, the number that each side of the equation $rac{3}{4}x = 9$ should be multiplied by to produce the equivalent equation of $x = 12$ is $rac{4}{3}$ . Looking back at the options provided:

A. $- rac{3}{4}$ (Incorrect - wrong sign and not reciprocal) B. $rac{1}{4}$ (Incorrect - not the reciprocal) C. $rac{4}{3}$ (Correct - this is the reciprocal of $rac{3}{4}$ ) D. 4 (Incorrect - this would require an additional step)

So, the answer is C. $rac{4}{3}$ . Mastering this technique is crucial for your algebra journey, guys!