Solving Equations: Kira's Substitution Method
Hey math enthusiasts! Let's dive into a cool problem where we get to flex our algebra muscles. We're looking at a system of equations, and we're going to see how Kira tackled it using the substitution method. This method is super handy for solving these kinds of problems, and understanding it will make your math life a whole lot easier, trust me. So, let's break down the problem step-by-step and see what Kira did. We have two equations, and our mission is to figure out the resulting equation after Kira's clever moves. We're going to uncover how she took the first equation, isolated a variable, and then used that information to simplify the second equation. It's like a puzzle, and we're the detectives figuring out the solution. Ready to get started, guys? Let's go!
Understanding the Problem: The System of Equations
Alright, first things first, let's get acquainted with the system of equations we're dealing with. It's like having two clues to solve a mystery. We have two equations, and both of them involve the variables x and y. Our goal is to find values for x and y that satisfy both equations simultaneously. Think of it like this: the solution to the system is a point (x, y) that lies on the graphs of both equations. So, the equations we're dealing with are:
- 3y = 12x
- x² + y² = 81
The first equation, 3y = 12x, is a linear equation. If you were to graph it, you'd get a straight line. The second equation, x² + y² = 81, is a quadratic equation, specifically the equation of a circle with a radius of 9 centered at the origin (0,0). What Kira did was to use one equation to simplify the other. That is the core idea of the substitution method. Now, let's see how Kira used these equations and what she did to find the resulting equation. Keep in mind that the resulting equation will involve only one variable. Cool, right?
Breaking Down the Equations
To make things easier, let's take a closer look at each equation. The first equation, 3y = 12x, is a linear equation. It expresses a direct relationship between x and y. We can actually simplify this equation further by dividing both sides by 3. This will help us isolate the variable y. The second equation, x² + y² = 81, represents all the points that are a distance of 9 units away from the origin. This equation is more complex because it involves squares of the variables. Kira's strategy, the substitution method, is perfect for dealing with situations like this. It allows us to reduce a system of equations into a simpler equation by replacing one of the variables. This substitution approach is often easier than trying to solve the equations directly, especially when we have a mix of linear and quadratic equations like these. It's like having a secret weapon in our mathematical arsenal.
Kira's Strategy: Isolating y and Substituting
Here’s where Kira’s brilliance comes in! Her first move was to isolate the variable y in the first equation. This means she rearranged the equation 3y = 12x so that y was all by itself on one side of the equation. This is a common and super useful technique in algebra. Once she did that, she had an expression for y in terms of x. Basically, she found out what y equals. To isolate y, we simply divide both sides of the equation 3y = 12x by 3:
y = (12x) / 3 y = 4x
So, y equals 4x. This is the key to the substitution method. Now, instead of y, we can write 4x. Next, she took this expression and substituted it into the second equation, x² + y² = 81. This means she replaced every instance of y in the second equation with 4x. This simple move transformed the second equation into one that only had x as a variable. It's like magically transforming the equation. This turns the second equation into a single-variable equation, which is way easier to handle. Now, we're ready for the grand finale: figuring out what this transformed equation looks like.
The Substitution Step-by-Step
Okay, let's get into the nitty-gritty of the substitution. We know that y = 4x. Now, let's substitute this value into the second equation, x² + y² = 81. Replacing y with 4x, the second equation becomes:
x² + (4x)² = 81
See how we've eliminated y and now have an equation only in terms of x? That's the power of substitution! We're not quite done yet, though. We need to simplify the equation to find out what it really looks like. Remember, our goal is to get the resulting equation that Kira came up with. Let's simplify and see how it goes.
The Resulting Equation: Unveiling the Answer
Alright, we're at the final step! We've isolated y and substituted it into the second equation. Now, we need to simplify the resulting equation to see what it is. We have x² + (4x)² = 81. First, let's simplify the (4x)² term. Remember that (4x)² means (4x) * (4x) which equals 16x². So, our equation becomes:
x² + 16x² = 81
Now, we can combine the like terms, which are x² and 16x². Adding them together, we get 17x². So, the equation simplifies to:
17x² = 81
And there you have it, guys! The resulting equation is 17x² = 81. This is the equation that Kira came up with after substituting the value of y from the first equation into the second equation. This equation is now ready to be solved for x. After solving for x, we can substitute the values of x back into the equation y = 4x to find the corresponding values of y. Therefore, the substitution method is a powerful tool to solve systems of equations, especially when we have a mix of linear and quadratic equations. It simplifies the problem and makes it easier to find the solution. Wasn't that fun?
Key Takeaways and Conclusion
Let's recap what we've learned and highlight the key takeaways. We started with a system of two equations and used the substitution method to solve it. Kira's clever move of isolating y and substituting it into the second equation transformed the problem into a simpler one. We found that the resulting equation is 17x² = 81. This equation is now a single-variable equation that we can solve to find the values of x. This method is extremely useful when dealing with equations that are difficult to solve directly. The substitution method allows us to simplify the system and make the process more manageable. Next time, if you face a similar problem, remember Kira's strategy: isolate a variable, substitute, and simplify. You'll be well on your way to solving the system. And that's a wrap! Hope you enjoyed the lesson. Keep practicing and exploring, and you'll become a math wizard in no time. See ya!