Solving For Y In The System 5x + 4y = -1 And Y = -6/4

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In the realm of mathematics, particularly in algebra, solving systems of equations is a fundamental skill. These systems often represent real-world scenarios where multiple variables interact, and finding the values of these variables is crucial for understanding and predicting outcomes. In this comprehensive guide, we will delve into the process of solving a specific system of equations for the variable y. This system is presented as follows:

5x + 4y = -1
y = -6/4

Our goal is to determine the value of y that satisfies both equations simultaneously. This involves a combination of algebraic techniques and a clear understanding of equation manipulation. Let's embark on this mathematical journey, breaking down each step and providing insights along the way.

Understanding the System of Equations

Before we dive into the solution, it's essential to grasp the nature of the system we're dealing with. We have two equations, each representing a relationship between two variables, x and y. The first equation, 5x + 4y = -1, is a linear equation, meaning its graph would be a straight line. The second equation, y = -6/4, is even simpler; it directly tells us the value of y. This is a significant advantage, as it allows us to bypass more complex methods like substitution or elimination, which are typically used when both equations contain both variables.

The second equation, y = -6/4, is particularly straightforward. It explicitly defines the value of y. This is a crucial piece of information that significantly simplifies the problem. We can directly use this value to understand the solution to the system. Furthermore, we can simplify the fraction -6/4 to -3/2, making the value of y even clearer. This simplification is always a good practice in mathematics, as it often makes subsequent calculations easier and the final answer more presentable.

Now that we have a clear understanding of the system, let's move on to the solution process. The fact that we already have a value for y makes this a relatively simple problem. However, it's still important to approach it systematically to ensure accuracy and to solidify our understanding of the underlying principles. We will explore how this direct knowledge of y can be used to potentially find the value of x as well, further illustrating the interconnectedness of variables in a system of equations.

Solving for y: The Direct Approach

The beauty of this particular problem lies in its simplicity. The second equation, y = -6/4, directly provides the solution for y. There's no need for complex algebraic manipulations or substitutions. We can simply state that the value of y is -6/4. However, it's always best practice to simplify fractions, so we can reduce -6/4 to -3/2. Therefore, the solution for y is:

y = -3/2

This direct approach highlights a fundamental principle in problem-solving: always look for the most straightforward path. In this case, the equation explicitly gave us the answer, saving us from unnecessary steps. However, this doesn't mean we should stop here. While we've solved for y, we can also use this information to find the value of x in the first equation. This will give us a complete solution to the system of equations.

Finding the value of x is a valuable exercise, as it reinforces our understanding of how variables interact within a system. It also demonstrates the power of substitution, a technique that is widely used in more complex equation-solving scenarios. By substituting the value we found for y into the first equation, we can transform it into an equation with only one variable, x, which we can then solve using basic algebraic operations.

This process of finding x not only completes the solution to the system but also provides a deeper understanding of the relationship between the two equations. It shows how the value of one variable can influence the value of another, a key concept in many mathematical and real-world applications. So, let's proceed to find the value of x, building upon our solution for y.

Finding x: Substitution in Action

Now that we've determined the value of y to be -3/2, we can substitute this value into the first equation, 5x + 4y = -1, to solve for x. This substitution is a crucial step, as it transforms the equation into one with a single variable, making it solvable. Let's perform the substitution:

5x + 4*(-3/2) = -1

Next, we simplify the equation:

5x - 6 = -1

Now, we isolate the term with x by adding 6 to both sides of the equation:

5x = 5

Finally, we solve for x by dividing both sides by 5:

x = 1

Therefore, we have found the value of x to be 1. This process demonstrates the power of substitution in solving systems of equations. By using the value of y that we previously determined, we were able to transform the original equation into a simpler one that we could easily solve for x.

This complete solution, with both x and y values, provides a comprehensive understanding of the system of equations. It's not just about finding the individual values; it's about understanding how these values interact and satisfy the given conditions. In this case, the values x = 1 and y = -3/2 simultaneously satisfy both equations in the system.

In the next section, we will verify our solution to ensure its accuracy and further solidify our understanding of the problem-solving process. Verification is an essential step in mathematics, as it helps us catch any potential errors and build confidence in our results.

Verifying the Solution

In mathematics, verification is a crucial step to ensure the accuracy of our solutions. It involves plugging the values we've found back into the original equations to see if they hold true. This process helps us catch any potential errors in our calculations and provides confidence in our results. We found that x = 1 and y = -3/2. Let's substitute these values into the original equations:

Equation 1: 5x + 4y = -1

Substituting x = 1 and y = -3/2, we get:

5*(1) + 4*(-3/2) = -1

Simplifying:

5 - 6 = -1

-1 = -1

The equation holds true.

Equation 2: y = -6/4

We already know that y = -3/2, which is the simplified form of -6/4. So, this equation also holds true.

Since both equations are satisfied by our values of x and y, we can confidently say that our solution is correct. This verification step underscores the importance of checking our work, especially in mathematics, where a small error can lead to a wrong answer.

By verifying our solution, we not only confirm its accuracy but also reinforce our understanding of the problem-solving process. It's a final check that ensures we've followed the correct steps and arrived at the correct answer. In the conclusion, we will summarize the steps we've taken to solve this system of equations and highlight the key takeaways from this exercise.

Conclusion: Mastering Systems of Equations

In this guide, we've successfully navigated the process of solving a system of equations for the variable y. We started with the system:

5x + 4y = -1
y = -6/4

We recognized the straightforward nature of the second equation, which directly provided the value of y. By simplifying -6/4 to -3/2, we found:

y = -3/2

We then leveraged this value to find x by substituting y = -3/2 into the first equation. This allowed us to transform the equation into one with a single variable, which we easily solved to find:

x = 1

Finally, we verified our solution by plugging both x and y values back into the original equations, confirming their accuracy.

This exercise demonstrates a fundamental approach to solving systems of equations: identifying the simplest path, utilizing direct information, and employing substitution to find unknown variables. It also highlights the importance of verification in ensuring the correctness of our solutions.

Solving systems of equations is a crucial skill in mathematics, with applications ranging from basic algebra to advanced calculus and beyond. By mastering these techniques, we equip ourselves with the tools to tackle a wide range of problems in various fields. The key takeaways from this guide include:

  • Always look for the simplest approach: In this case, the second equation directly provided the value of y, saving us from more complex methods.
  • Simplify fractions: Reducing fractions like -6/4 to -3/2 makes calculations easier and the answer more presentable.
  • Substitution is a powerful tool: Substituting known values into equations can transform them into simpler forms that are easier to solve.
  • Verification is essential: Always check your solutions to ensure their accuracy.

By applying these principles, you can confidently approach and solve a wide variety of systems of equations. Remember, practice is key to mastering any mathematical skill, so continue to challenge yourself with new problems and explore different techniques.