Solving For Q A Step-by-Step Guide To Solving Q + 5/9 = 5/6
Hey guys! Today, we're diving into a super common type of math problem: solving for a variable. In this case, we're tackling the equation q + 5/9 = 5/6. Don't worry if it looks a little intimidating at first. We're going to break it down step-by-step, so it'll be a piece of cake by the end. Think of it like this, we're on a mission to isolate 'q' all by itself on one side of the equation. Once we do that, we've found our answer. So, grab your pencils, and let's get started!
Understanding the Equation
Before we jump into solving, let's make sure we understand what the equation is telling us. The equation q + 5/9 = 5/6 is an algebraic statement that says "some number, which we're calling 'q', plus five-ninths is equal to five-sixths." Our goal is to find the exact value of 'q' that makes this statement true. Essentially, we need to figure out what number, when added to 5/9, gives us 5/6. This might seem a bit abstract, but it's a fundamental concept in algebra. We use variables like 'q' to represent unknown quantities, and equations to show relationships between those quantities. By solving equations, we can unlock the values of these unknowns and solve all sorts of problems in math, science, and even everyday life.
The key to solving any equation is to remember the golden rule: what you do to one side of the equation, you must do to the other. This is crucial for maintaining the balance and ensuring that the equation remains true. If we add something to the left side, we have to add the same thing to the right side. If we multiply the right side by a number, we must multiply the left side by the same number. This principle will guide us as we isolate 'q'. So, with this understanding in place, let's move on to the actual steps involved in solving for 'q'. We'll start by figuring out how to get rid of that pesky 5/9 that's hanging out with our 'q'.
Step 1 Isolating 'q'
The first crucial step in solving the equation q + 5/9 = 5/6 is to isolate 'q'. Remember, our mission is to get 'q' all by itself on one side of the equation. Currently, we have 'q' plus 5/9. To get rid of the 5/9, we need to perform the opposite operation. Since we're adding 5/9, the opposite operation is subtraction. So, we're going to subtract 5/9 from both sides of the equation. This is where that golden rule we talked about earlier comes into play – whatever we do to one side, we must do to the other to keep the equation balanced.
So, let's write that down mathematically: q + 5/9 - 5/9 = 5/6 - 5/9. Notice that we've subtracted 5/9 from both sides. On the left side, the +5/9 and -5/9 cancel each other out, leaving us with just 'q'. This is exactly what we wanted! Now our equation looks like this: q = 5/6 - 5/9. We've successfully isolated 'q' on the left side. The next step is to simplify the right side of the equation, which involves subtracting the two fractions. This means we need to find a common denominator, which will allow us to combine the fractions into a single value. Don't worry, we'll walk through this process step-by-step in the next section.
Step 2 Finding a Common Denominator
Now that we've isolated 'q' and have the equation q = 5/6 - 5/9, the next step is to subtract the fractions on the right side. But, we can't just subtract fractions directly unless they have the same denominator. This is because the denominator tells us how many equal parts the whole is divided into. To subtract them, we need to express both fractions in terms of the same "size" parts. That's where the concept of a common denominator comes in. A common denominator is simply a number that both denominators (in this case, 6 and 9) divide into evenly.
So, how do we find a common denominator? One way is to list out the multiples of each denominator and look for the smallest multiple they share. The multiples of 6 are: 6, 12, 18, 24, 30, and so on. The multiples of 9 are: 9, 18, 27, 36, and so on. Notice that the smallest multiple they have in common is 18. This is called the least common multiple (LCM), and it will be our common denominator. Alternatively, you can find the least common multiple by prime factorization. The prime factors of 6 are 2 and 3 (6 = 2 x 3). The prime factors of 9 are 3 and 3 (9 = 3 x 3). To find the LCM, we take the highest power of each prime factor present in either number: 2 x 3 x 3 = 18. So, we've confirmed that 18 is indeed our common denominator. Now we need to rewrite our fractions with this new denominator.
Step 3 Rewriting Fractions
Now that we've found our common denominator, which is 18, we need to rewrite the fractions 5/6 and 5/9 so that they both have a denominator of 18. This involves multiplying both the numerator and the denominator of each fraction by a suitable number. Remember, we're essentially multiplying the fraction by a form of 1, which doesn't change its value, only its appearance. Let's start with 5/6. We need to figure out what number we can multiply 6 by to get 18. Well, 6 times 3 is 18. So, we'll multiply both the numerator and the denominator of 5/6 by 3: (5/6) * (3/3) = 15/18. So, 5/6 is equivalent to 15/18.
Now, let's do the same for 5/9. We need to figure out what number we can multiply 9 by to get 18. Well, 9 times 2 is 18. So, we'll multiply both the numerator and the denominator of 5/9 by 2: (5/9) * (2/2) = 10/18. So, 5/9 is equivalent to 10/18. Now we've successfully rewritten both fractions with the common denominator of 18. Our equation now looks like this: q = 15/18 - 10/18. We're one step closer to finding the value of 'q'! The next step is to actually perform the subtraction, which is much easier now that the fractions have the same denominator.
Step 4 Subtracting Fractions
With our fractions rewritten to have a common denominator, we're ready to subtract! Our equation currently looks like this: q = 15/18 - 10/18. When subtracting fractions with a common denominator, we simply subtract the numerators and keep the denominator the same. So, 15/18 minus 10/18 is (15 - 10)/18. Performing the subtraction in the numerator, we get 5/18. Therefore, q = 5/18. We've done it! We've successfully isolated 'q' and simplified the right side of the equation to find its value. But, before we celebrate too much, it's always a good idea to check our answer to make sure it's correct.
Step 5 Checking the Solution
Alright, we think we've found the solution: q = 5/18. But, to be absolutely sure, we need to check our answer. The best way to do this is to plug our solution back into the original equation and see if it makes the equation true. Our original equation was q + 5/9 = 5/6. Let's substitute 5/18 for 'q': 5/18 + 5/9 = 5/6. Now we need to see if the left side of the equation actually equals the right side. To add 5/18 and 5/9, we need a common denominator, which we already know is 18. We need to rewrite 5/9 with a denominator of 18. We already did this earlier, but let's do it again to refresh our memory: (5/9) * (2/2) = 10/18. So, our equation now looks like this: 5/18 + 10/18 = 5/6. Adding the fractions on the left side, we get: (5 + 10)/18 = 15/18. So, we have: 15/18 = 5/6.
Now, let's simplify 15/18 by dividing both the numerator and denominator by their greatest common factor, which is 3: (15/3) / (18/3) = 5/6. So, we have: 5/6 = 5/6. Hooray! The left side equals the right side. This means our solution, q = 5/18, is correct! We've successfully solved the equation. Give yourself a pat on the back!
Conclusion
So there you have it, guys! We've successfully solved the equation q + 5/9 = 5/6. We walked through each step, from isolating 'q' to finding a common denominator, rewriting fractions, subtracting, and finally, checking our solution. Remember, the key to solving equations is to understand the underlying principles and to take it one step at a time. Don't be afraid to break down the problem into smaller, more manageable parts. With practice, you'll become a pro at solving for variables. The most important takeaway is the concept of keeping the equation balanced – what you do to one side, you must do to the other. This ensures that your solution remains accurate. Keep practicing, and you'll master these skills in no time! Now you can confidently tackle similar equations and impress your friends and teachers with your math prowess. Keep up the great work!
