Solving Systems Of Equations 10x^2 - Y = 48 And 2y = 16x^2 + 48

Hey guys! Today, we're diving into a super interesting problem from the world of mathematics – solving systems of equations. Specifically, we're going to tackle a system involving quadratic equations. Don't worry if that sounds intimidating; we'll break it down step by step, making sure everyone understands the process. So, let's jump right into it!

The Challenge: Deciphering the System of Equations

Our main focus is to figure out solutions of systems of equations and we've got this system staring back at us:

\begin{cases}
10x^2 - y = 48 \\
2y = 16x^2 + 48
\end{cases}

This looks a bit complex, right? We've got two equations, two unknowns (x and y), and a bunch of squares and numbers. The challenge is to find the values of x and y that satisfy both equations simultaneously. These values, when plugged into both equations, will make both statements true. Think of it like finding the perfect matching pieces for a puzzle. There are a couple of approaches we can take, and we're going to explore the most efficient one here.

Understanding the Equations: A Closer Look

Before we start crunching numbers, let's get a feel for what these equations represent. The first equation, 10x² - y = 48, tells us about a relationship between x² and y. It says that if you take 10 times the square of x and subtract y, you should end up with 48. The second equation, 2y = 16x² + 48, gives us another relationship. It says that twice y is equal to 16 times the square of x, plus 48. Notice that both equations involve x², which is a key observation that will help us in solving the system.

Strategic Moves: Planning Our Attack

When faced with a system like this, it's crucial to have a plan. We could try substitution, where we solve one equation for one variable and plug it into the other equation. Or, we could try elimination, where we manipulate the equations to eliminate one variable. In this case, elimination looks promising because we have x² terms in both equations. Our goal will be to manipulate the equations so that either the x² terms or the y terms cancel out when we add or subtract the equations. This will leave us with a simpler equation in just one variable, which we can then easily solve. Once we have the value of one variable, we can plug it back into either of the original equations to find the value of the other variable.

The Elimination Method: A Step-by-Step Solution

Let's tackle this system of equations using the elimination method. This method is particularly effective when we can easily manipulate the equations to cancel out one of the variables. Here's how we'll do it:

Step 1: Setting the Stage for Elimination

The first thing we want to do is make sure the coefficients of either x² or y are the same (or opposites) in both equations. Looking at our system:

\begin{cases}
10x^2 - y = 48 \\
2y = 16x^2 + 48
\end{cases}

We notice that the y terms have coefficients of -1 and 2. To make these opposites, we can multiply the first equation by 2. This gives us:

2 * (10x^2 - y) = 2 * 48

Which simplifies to:

20x^2 - 2y = 96

Now our system looks like this:

\begin{cases}
20x^2 - 2y = 96 \\
2y = 16x^2 + 48
\end{cases}

Step 2: Eliminating the Variable

Now we're in a perfect position to eliminate y. Notice that the y terms are -2y and 2y. If we add the two equations together, these terms will cancel out:

(20x^2 - 2y) + (2y) = 96 + (16x^2 + 48)

Simplifying, we get:

20x^2 = 16x^2 + 144

Step 3: Solving for x

Now we have a single equation with just x². Let's solve for x²:

20x^2 - 16x^2 = 144

4x^2 = 144

x^2 = 36

Taking the square root of both sides, we get two possible values for x:

x = ±6

So, x can be either 6 or -6. This is a crucial step, as it shows us that we'll likely have two sets of solutions for our system.

Step 4: Finding the Corresponding y Values

Now that we have the x values, we need to find the corresponding y values. We can plug each x value back into either of the original equations. Let's use the second equation, 2y = 16x² + 48, as it looks a bit simpler.

Case 1: x = 6

Plugging x = 6 into the equation, we get:

2y = 16(6)^2 + 48

2y = 16 * 36 + 48

2y = 576 + 48

2y = 624

y = 312

So, one solution is (6, 312).

Case 2: x = -6

Now, let's plug in x = -6:

2y = 16(-6)^2 + 48

2y = 16 * 36 + 48

2y = 576 + 48

2y = 624

y = 312

Interestingly, we get the same y value, y = 312, for x = -6. So, our second solution is (-6, 312).

Step 5: The Grand Finale – The Solutions

We've done it! We've successfully navigated the system of equations and found our solutions. The solutions to the system are:

• (6, 312)
• (-6, 312)

These are the two points where the graphs of the two equations would intersect, if we were to visualize them. It's always a good idea to check our solutions by plugging them back into the original equations to make sure they hold true. Let's quickly verify:

Checking (6, 312):

• Equation 1: 10(6)² - 312 = 10 * 36 - 312 = 360 - 312 = 48 (Correct!)
• Equation 2: 2(312) = 16(6)² + 48 = 624 = 16 * 36 + 48 = 576 + 48 = 624 (Correct!)

Checking (-6, 312):

• Equation 1: 10(-6)² - 312 = 10 * 36 - 312 = 360 - 312 = 48 (Correct!)
• Equation 2: 2(312) = 16(-6)² + 48 = 624 = 16 * 36 + 48 = 576 + 48 = 624 (Correct!)

Both solutions check out, so we can confidently say we've cracked the code!

Alternative Approaches: Substitution Method

While we successfully used the elimination method, it's worth noting that we could also have used the substitution method. Let's briefly explore how that would work.

Substitution Strategy

In the substitution method, we solve one equation for one variable and then substitute that expression into the other equation. Looking at our original system:

\begin{cases}
10x^2 - y = 48 \\
2y = 16x^2 + 48
\end{cases}

We could solve the first equation for y:

y = 10x^2 - 48

Then, we would substitute this expression for y into the second equation:

2(10x^2 - 48) = 16x^2 + 48

The Substitution Process

Let's continue with the substitution:

20x^2 - 96 = 16x^2 + 48

Now, we solve for x²:

20x^2 - 16x^2 = 48 + 96

4x^2 = 144

x^2 = 36

Which gives us the same x values as before:

x = ±6

Finding y via Substitution

Now we plug these x values back into our expression for y:

y = 10x^2 - 48

Case 1: x = 6

y = 10(6)^2 - 48

y = 10 * 36 - 48

y = 360 - 48

y = 312

Case 2: x = -6

y = 10(-6)^2 - 48

y = 10 * 36 - 48

y = 360 - 48

y = 312

As we can see, the substitution method leads us to the same solutions: (6, 312) and (-6, 312). This reinforces the idea that there are often multiple paths to the same destination in mathematics!

Key Takeaways: Mastering Systems of Equations

Solving systems of equations might seem daunting at first, but with a systematic approach and a bit of practice, it becomes much more manageable. Here are some key takeaways from our journey today:

1. Understanding the Problem: Before diving into calculations, take a moment to understand the equations and what they represent. Look for patterns and relationships between the variables.
2. Choosing the Right Method: Decide whether elimination or substitution is the more efficient method for the given system. Consider the structure of the equations and which method will lead to the simplest calculations. The elimination method is especially useful when the coefficients of one of the variables are the same or easily made the same. The substitution method works best when one equation can be easily solved for one variable.
3. Step-by-Step Approach: Break the problem down into smaller, manageable steps. This will help you stay organized and avoid errors. Follow the steps carefully, whether you're eliminating a variable or substituting an expression.
4. Double-Check Your Solutions: Always verify your solutions by plugging them back into the original equations. This ensures that your answers are correct and satisfies the entire system. Checking solutions is a crucial step in solving any mathematical problem.
5. Multiple Solution Paths: Remember that there may be more than one way to solve a problem. If one method seems difficult, try another. Understanding different methods gives you flexibility and a deeper understanding of the underlying concepts.

Real-World Applications: Why This Matters

You might be wondering,