Solving Proportions: Find K In 9/(k-7) = 6/k

Hey guys! Today, we're diving into the world of proportions and tackling a fun little problem. We're going to figure out how to solve for k in the proportion 9/(k-7) = 6/k. It might seem a bit tricky at first, but trust me, we'll break it down step by step so it’s super easy to understand. So, grab your pencils, and let's get started!

Understanding Proportions

Before we jump into solving our specific problem, let's make sure we're all on the same page about what proportions actually are. At their heart, proportions are simply statements that two ratios are equal. Think of a ratio as a comparison between two quantities. For example, if you have 9 apples and k-7 oranges, the ratio of apples to oranges is 9:(k-7). When we set two of these ratios equal to each other, we get a proportion.

In mathematical terms, a proportion looks like this: a/b = c/d. This means the ratio of a to b is the same as the ratio of c to d. Proportions pop up everywhere in real life, from scaling recipes to calculating distances on a map. So, mastering them is a seriously useful skill.

Why are Proportions Important?

Understanding proportions isn't just about acing math tests; it's a skill that translates to tons of real-world situations. Need to double a recipe? You're using proportions! Planning a road trip and figuring out how long it will take? Proportions again! They help us scale things up or down, compare different quantities, and make predictions based on existing data.

For example, let's say you're baking a cake, and the recipe calls for 2 eggs for every cup of flour. If you want to make a bigger cake that needs 3 cups of flour, you can use a proportion to figure out how many eggs you'll need. The proportion would look something like this: 2 eggs / 1 cup flour = x eggs / 3 cups flour. Solving for x will tell you exactly how many eggs you need. Pretty neat, huh?

Setting up the Proportion: 9/(k-7) = 6/k

Alright, now that we've got a solid grasp on what proportions are, let's zoom in on our specific problem: 9/(k-7) = 6/k. The first thing we need to do is identify what each part of the proportion represents. We have two ratios here: 9 is being compared to (k-7), and 6 is being compared to k. Our mission is to find the value of k that makes these two ratios truly equal.

Identifying the Parts

In our proportion 9/(k-7) = 6/k:

• 9 and 6 are the numerators (the top numbers) of our fractions.
• (k-7) and k are the denominators (the bottom numbers) of our fractions.
• The equals sign (=) tells us that the ratio on the left (9/(k-7)) is the same as the ratio on the right (6/k).

Think of it like this: We're saying that 9 divided by some quantity (k-7) gives us the same result as 6 divided by k. The key is to figure out what that mysterious k is!

The Cross-Multiplication Method

So, how do we actually solve for k? The most common technique for tackling proportions is something called cross-multiplication. This is a super handy trick that simplifies our equation and makes it much easier to work with. The basic idea behind cross-multiplication is that if a/b = c/d, then ad = bc. In other words, we multiply the numerator of the first fraction by the denominator of the second, and set that equal to the product of the denominator of the first fraction and the numerator of the second.

Applying Cross-Multiplication

Let's apply this to our proportion, 9/(k-7) = 6/k. Following the cross-multiplication rule, we get:

9 * k = 6 * (k - 7)

See what we did there? We multiplied 9 by k and set it equal to 6 multiplied by the entire expression (k - 7). This step is crucial because it transforms our proportion into a more manageable equation.

Now, we have a linear equation that we can solve using basic algebraic techniques. The next step involves distributing the 6 on the right side of the equation. This means we multiply 6 by both terms inside the parentheses: k and -7.

Distributing and Simplifying

Okay, let's take our equation from the previous step:

9k = 6 * (k - 7)

Now, we need to distribute the 6 on the right side. This means multiplying 6 by both k and -7:

9k = 6k - 42

See how we multiplied 6 by k to get 6k and 6 by -7 to get -42? This step is super important for simplifying the equation and getting us closer to solving for k.

Why Distribution Matters

The distributive property is a fundamental concept in algebra. It allows us to handle expressions that involve parentheses by multiplying the term outside the parentheses by each term inside. If we skipped this step, we'd be stuck with a much more complicated equation that's harder to solve. So, remember to always distribute when you see parentheses in your equations!

Now that we've distributed, our equation looks like this:

9k = 6k - 42

The next step is to isolate the terms with k on one side of the equation. We can do this by subtracting 6k from both sides.

Isolating the Variable

Our current equation is:

9k = 6k - 42

To get all the k terms on one side, we're going to subtract 6k from both sides of the equation. Remember, whatever we do to one side, we have to do to the other to keep the equation balanced. This is a golden rule in algebra!

So, let's subtract 6k from both sides:

9k - 6k = 6k - 42 - 6k

Now, simplify both sides:

3k = -42

Look at that! We've successfully isolated the k term. We're almost there! We now have a much simpler equation: 3k = -42. The only thing left to do is to get k all by itself. How do we do that? By dividing both sides by 3, of course!

Solving for k

We've reached the final step! Our equation is now:

3k = -42

To solve for k, we need to get it all alone on one side of the equation. Since k is being multiplied by 3, we need to do the opposite operation: divide both sides by 3.

So, let's divide both sides by 3:

3k / 3 = -42 / 3

Now, simplify:

k = -14

And there you have it! We've found the value of k. It's -14. 🎉

Checking Our Solution

It's always a good idea to double-check your answer, especially in math problems. To do this, we'll plug our value of k (-14) back into the original proportion and see if it holds true.

Our original proportion was:

9 / (k - 7) = 6 / k

Now, substitute k = -14:

9 / (-14 - 7) = 6 / -14

Simplify the left side:

9 / (-21) = 6 / -14

Reduce both fractions:

-3 / 7 = -3 / 7

Yep, it checks out! Both sides are equal, so we know that k = -14 is indeed the correct solution.

Conclusion

Woo-hoo! We did it! We successfully solved the proportion 9/(k-7) = 6/k and found that k = -14. We walked through each step, from understanding what proportions are to using cross-multiplication, distributing, isolating the variable, and finally, solving for k. And, just to be sure, we even checked our answer.

Remember, guys, proportions are powerful tools that can help us in all sorts of situations. So, keep practicing, and you'll become a proportion pro in no time! If you ever get stuck, just break the problem down step by step, and don't forget the magic of cross-multiplication. You've got this!