Equation With Proportionality Constant Of 9?
Hey there, math enthusiasts! Today, we're diving into the fascinating world of proportionality and exploring how to identify equations with a specific constant of proportionality. This concept is fundamental in various areas of mathematics and science, so let's break it down in a way that's both easy to understand and engaging.
Understanding Constant of Proportionality
Before we jump into solving the problem, let's ensure we're all on the same page regarding what the constant of proportionality actually means. In a nutshell, it's the constant value that relates two proportional quantities. Think of it as the magical number that connects two variables in a special relationship. Specifically, when we talk about two variables, usually x and y, being proportional, we mean that their ratio is always constant. This can be expressed mathematically as:
y = kx
Where:
- y and x are the two variables.
- k is the constant of proportionality. This is the key player we're focusing on today!
So, what does this k really tell us? Well, it tells us how many units of y we get for every one unit of x. If k is a large number, y changes rapidly with x. If k is small, y changes more gradually. If k is 9, then it simply implies that y is nine times the value of x. In essence, the constant of proportionality acts as the multiplier that scales x to give you y. In simpler terms, k is the number you multiply x by to get y. Let’s dive in and examine some examples to make this concept even clearer.
Consider a scenario where you're buying apples at a store. The total cost (y) is proportional to the number of apples you buy (x). If each apple costs $0.50, then the equation representing this relationship is:
y = 0.50x
Here, the constant of proportionality (k) is 0.50. This means for every apple you buy, the cost increases by $0.50. Similarly, if you’re converting meters to centimeters, the relationship is:
centimeters = 100 * meters
The constant of proportionality is 100, because there are 100 centimeters in every meter.
Understanding the concept of the constant of proportionality is so important because it pops up all over the place in math and science. From calculating distances based on speed and time to understanding how quantities change in chemistry or physics, this concept is a fundamental building block. It allows us to create mathematical models that accurately describe relationships between different quantities, making it easier to predict outcomes and solve problems. So, now that we have a solid grip on what the constant of proportionality is, let's get back to our original question and find the equation where this constant equals 9. It's going to be a breeze!
Analyzing the Given Equations
Now, let's carefully examine the given equations and identify which one has a constant of proportionality of 9. Remember, we're looking for an equation in the form y = kx, where k is the magic number 9.
Here are the equations we need to analyze:
A. $y=\frac{81}{3} x$ B. $y=9 x$ C. $y=3 x$ D. $y=\frac{1}{9} x$
Let's break down each option step by step:
Option A: $y=\frac{81}{3} x$
In this equation, we see a fraction multiplied by x. To make things clearer, let's simplify the fraction. 81 divided by 3 equals 27. So, we can rewrite the equation as:
y = 27x
Now, it's much easier to see the constant of proportionality. It's 27, which means for every unit increase in x, y increases by 27 units. But wait, we're looking for a constant of 9, so this option doesn't fit the bill. It’s close, but not quite the bullseye!
Option B: $y=9 x$
This equation looks promising! It's already in the standard form y = kx. Can you spot the constant of proportionality? It's sitting right there in front of the x! The number multiplying x is 9. This means that for every one unit increase in x, y increases by 9 units. That’s exactly what we’re looking for! So, this option looks like a strong contender. But, let's be thorough and check the remaining options just to be sure.
Option C: $y=3 x$
In this equation, the number multiplying x is 3. This means that for every unit increase in x, y increases by 3 units. So, the constant of proportionality here is 3. It’s a simple proportional relationship, but it's not the 9 we're after. This option is out of the running.
Option D: $\frac{1}{9} x$
Here, the number multiplying x is the fraction 1/9. This means that for every unit increase in x, y increases by only 1/9 of a unit. So, the constant of proportionality in this case is 1/9, a fraction much smaller than 9. This option clearly isn't what we're looking for.
By carefully analyzing each equation, we’ve pinpointed the one that matches our requirement. Let’s move on to confirming our answer.
Confirming the Correct Answer
Alright, we've dissected each equation and identified a prime candidate. Let's recap what we've found. We were on the hunt for an equation with a constant of proportionality equal to 9. We examined four options:
A. $y=\frac{81}{3} x$ which simplifies to y = 27x B. $y=9 x$ C. $y=3 x$ D. $y=\frac{1}{9} x$
We simplified option A to y = 27x, making it clear that the constant is 27, not 9. Options C and D had constants of 3 and 1/9, respectively, so they didn't make the cut either. This leaves us with option B: $y=9x$.
In this equation, the constant of proportionality is indeed 9. For every one unit increase in x, y increases by 9 units. This perfectly matches the condition we were given. So, we can confidently say that option B is the correct answer!
To double-check, let’s plug in a simple value for x, like 1, into the equation. If x is 1, then y would be 9 * 1 = 9. This confirms that the relationship is indeed proportional with a constant of 9. If we tried this with the other equations, we'd see that they don't yield a proportional relationship with a constant of 9.
For example, in option A, if x is 1, y would be 27. In option C, if x is 1, y would be 3, and in option D, if x is 1, y would be 1/9. None of these match our target constant of 9.
Therefore, we can confidently conclude that option B is the equation with a constant of proportionality equal to 9. High five! We've successfully navigated this math challenge.
Final Answer
Therefore, the equation that has a constant of proportionality equal to 9 is:
B. $y=9 x$
