Solving Systems Of Equations: Find Y When X = 5

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Hey guys! Today, we're diving into the fascinating world of solving systems of equations. Specifically, we're going to tackle a problem where we need to find the value of 'y' given a system of two equations and the value of 'x'. Don't worry, it sounds more complicated than it is! We'll break it down step by step so it's super easy to follow. So, let's get started and become equation-solving pros!

Understanding Systems of Equations

Before we jump into the problem, let's make sure we're all on the same page about what a system of equations actually is. Think of it as a puzzle where you have multiple equations with multiple unknown variables (like 'x' and 'y'). Our goal is to find the values of these variables that make all the equations true at the same time. There are several methods to solve these systems, such as substitution, elimination, and graphing. Each method has its own strengths, and the best one to use often depends on the specific equations you're dealing with.

  • Why are systems of equations important? You might be wondering, “Why do we even need to learn this stuff?” Well, systems of equations pop up in all sorts of real-world situations! From figuring out the cost of different items when you know the total price and some relationships between them, to modeling complex relationships in science and engineering, these skills are incredibly useful. For instance, imagine you're planning a party and need to figure out how many pizzas and sodas to buy based on your budget and the number of guests. You could totally use a system of equations to solve that!
  • Different methods, different approaches: As mentioned earlier, there are a few key methods for tackling systems of equations. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This is great when one of the equations is already solved for a variable or can be easily rearranged. The elimination method, on the other hand, focuses on adding or subtracting the equations to eliminate one of the variables. This works best when the coefficients of one of the variables are the same or easily made the same. And then there's the graphing method, where you plot the equations on a graph and find the point of intersection, which represents the solution. This method is fantastic for visualizing the solution, but it might not be the most precise if the solution involves fractions or decimals.

In our case, we're given a specific value for 'x', which makes things a whole lot easier. We'll primarily use the substitution method here, but understanding the other methods gives you a broader toolkit for future problems!

The Problem at Hand

Okay, let's get down to the nitty-gritty! We have the following system of equations:

  1. 5x + 2y = -24
  2. 4x + 3y = -8

And we're given that x = 5. Our mission, should we choose to accept it, is to find the value of 'y'.

  • Why is this problem interesting? This problem is a classic example of how we can use given information to simplify a seemingly complex situation. Knowing the value of 'x' drastically reduces the amount of work we need to do. It's like having a cheat code for the equation-solving game! It also highlights the power of substitution, which is a fundamental technique in algebra and beyond.
  • Breaking down the challenge: At first glance, two equations with two unknowns might seem daunting. But the key here is that we're not starting from scratch. We already know what 'x' is. This means we can plug that value into our equations and turn them into simpler equations with only one unknown – 'y'. This is a common strategy in problem-solving: break down a complex problem into smaller, more manageable parts.
  • The importance of clear steps: When solving equations, it's super important to be organized and show your work. This not only helps you avoid mistakes but also makes it easier for others (and your future self!) to understand your thinking. We'll go through each step meticulously, explaining the reasoning behind each move. This way, you won't just learn the answer to this specific problem, but you'll also develop a solid process for tackling similar problems in the future.

So, with our problem clearly defined and our strategy in place, let's roll up our sleeves and get to work!

Step-by-Step Solution

Now comes the fun part – actually solving the problem! Here's how we'll do it, step by step:

Step 1: Substitute the value of x

Since we know that x = 5, we can substitute this value into both equations. This will give us two new equations that only involve 'y'.

  • Equation 1: 5x + 2y = -24 becomes 5(5) + 2y = -24
  • Equation 2: 4x + 3y = -8 becomes 4(5) + 3y = -8

This is a crucial step because it simplifies our problem significantly. We've effectively reduced two equations with two unknowns into two equations with just one unknown. That's progress!

Step 2: Simplify the equations

Now, let's simplify the equations by performing the multiplication:

  • Equation 1: 25 + 2y = -24
  • Equation 2: 20 + 3y = -8

We're getting closer! The equations are looking much cleaner now. Notice how the arithmetic is relatively straightforward here. This is often the case when you break down a problem into smaller steps. Each step becomes less intimidating.

Step 3: Solve for y in either equation

We can choose either equation to solve for 'y'. Let's start with Equation 1 (25 + 2y = -24). To isolate 'y', we'll first subtract 25 from both sides:

2y = -24 - 25 2y = -49

Now, divide both sides by 2:

y = -49 / 2 y = -24.5

We've found a value for 'y'! But to be absolutely sure, let's check our answer using the second equation.

Step 4: Verify the solution using the other equation

Let's plug y = -24.5 into Equation 2 (20 + 3y = -8) to see if it holds true:

20 + 3(-24.5) = -8 20 - 73.5 = -8 -53.5 = -8

Hmmm… something seems off. -53.5 does not equal -8. This means we might have made a mistake somewhere in our calculations. It's super important to check your work, guys! Let's go back and see if we can spot the error.

Step 5: Re-evaluate and correct the mistake

Okay, let's carefully go back through our steps. We substituted x = 5 correctly, and we simplified the equations correctly. The potential issue is in Step 3, where we solved for 'y' in Equation 1. Let's double-check that:

25 + 2y = -24 2y = -24 - 25 2y = -49 y = -49 / 2 y = -24.5

That looks correct… So, the problem must be when we verified the solution in Equation 2:

20 + 3(-24.5) = -8 20 - 73.5 = -8 -53.5 = -8

Ah, I see the mistake! We made an arithmetic error. While the calculations are correct until -53.5, this does not equal -8, indicating an issue. Let’s try solving Equation 2 (20 + 3y = -8) directly for y:

3y = -8 - 20 3y = -28 y = -28 / 3

Okay, this gives us a different value for 'y'. Let's use this new value of y = -28/3 to verify within equation 1:

5(5) + 2(-28/3) = -24 25 - 56/3 = -24 (75 - 56)/3 = -24 19/3 = -24

This is still incorrect meaning that, there might be an error with the initial conditions of the problem. The result in x = 5 is not yielding a valid solution when substituted back into the equations.

Step 6: Final Answer

Given the inconsistency with x = 5 when validating the solution for y across both equations, it suggests that either the provided x value or the system of equations themselves might not yield a consistent solution. It's crucial to re-evaluate the problem statement or the equations if such a discrepancy arises. However, if we were to solve equation 2 for y directly, assuming x = 5 is a valid condition within the scope of a potential problem setup issue:

From 4x + 3y = -8, with x = 5:

4(5) + 3y = -8

20 + 3y = -8

3y = -28

y = -28/3

Therefore, considering the context and the possible discrepancy, the calculated value for y is approximately -9.33, assuming there's an external condition or a need to find y based on the second equation alone given x = 5.

Final Answer: y = -28/3

Key Takeaways

  • Substitution is powerful: When you know the value of one variable, substitution is your best friend. It simplifies the problem and lets you solve for the other variable.
  • Always verify: Don't just stop when you get an answer! Plug it back into the original equations to make sure it works. This is a crucial step in avoiding errors.
  • Don't be afraid to re-evaluate: If your solution doesn't check out, don't get discouraged. Go back through your steps and look for potential mistakes. Problem-solving is a process of trial and error.
  • Be careful with arithmetic: Simple arithmetic errors can throw off your entire solution. Take your time and double-check your calculations.

Solving systems of equations can seem intimidating at first, but with practice and a clear, step-by-step approach, you'll become a pro in no time! Remember to break down the problem, use the information you're given, and always verify your answers. Keep practicing, and you'll be solving even the trickiest equations with confidence!

I hope this explanation helped you guys understand how to solve this type of problem. Keep practicing, and you'll become a math whiz in no time!