Solving Systems Of Equations: A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving into the fascinating world of solving systems of equations. Specifically, we'll tackle the system:

$\begin{array}{l} y=-3 x-2 \\ 5 x+2 y=15 \end{array}$

Don't worry, it might seem a bit daunting at first, but trust me, with a few simple steps, we'll crack this code together. So, grab your pencils, and let's get started! Our main goal is to find the values of x and y that satisfy both equations simultaneously. Think of it like finding a secret meeting point on a graph where the two lines intersect. There are several methods to achieve this, but we'll focus on the substitution method here, as it's particularly well-suited for this problem. The solution to the system of equations is more than just finding the answers; it's about understanding the relationships between the variables and how they interact to create a balanced equation. By the end of this article, you'll be equipped with the knowledge and confidence to tackle similar problems. So, let's break down each step in detail and unravel the mystery together. Remember, practice makes perfect, so be sure to try some examples on your own after we're done here. Let's start with the first equation which is already solved for y. This is a great starting point, as we can easily use this equation to substitute into the second equation. This method is a cornerstone in algebra, providing a direct approach to solving for unknown variables. Understanding how to solve such systems is fundamental in mathematics, paving the way for more complex problem-solving in areas like calculus, physics, and engineering. The importance of these skills cannot be overstated; they serve as a foundation for advanced studies and real-world applications. The substitution method allows us to transform a system of equations into a single equation with a single variable, making it easier to solve. This process exemplifies how mathematical techniques can simplify and solve intricate problems. We are going to go through this step by step.

Step 1: Substitution

Alright, folks, let's jump right into the meat of the matter! We're going to use the substitution method to solve this system. Since the first equation is already solved for y (y = -3x - 2), we're going to take that expression for y and plug it into the second equation. This is where the magic happens! We're essentially replacing y in the second equation with what it's equal to, based on the first equation. This will give us a new equation with only one variable, x, which we can then solve. This process is like simplifying a complex puzzle by breaking it down into smaller, manageable pieces. The strategy is straightforward but incredibly powerful, allowing us to reduce a multi-variable problem into a single-variable problem. This is a critical step in algebra, and understanding it will boost your confidence and problem-solving skills. Remember, the goal is to isolate the variable, which allows you to find a single, definitive answer. The key is to apply the substitution accurately and carefully, ensuring that you don't miss any terms or make any errors in your calculations. Taking the time to double-check each step will help you avoid common pitfalls. This methodical approach will not only help you find the correct solution but also improve your overall understanding of mathematical concepts. Understanding how to solve for each variable is important in algebra. Let's do it! So, here's how it looks:

• Original Second Equation: 5x + 2y = 15
• Substitute y: 5x + 2(-3x - 2) = 15

See? We've replaced y with its equivalent from the first equation. Now, we'll simplify and solve for x. Make sure to keep the equation balanced.

Step 2: Simplify and Solve for x

Now, let's do some algebraic housekeeping and simplify the equation we got in the previous step. We'll start by distributing the 2 across the terms inside the parentheses. This step is about expanding the equation and preparing it for isolation. Remember, the goal is to get x all by itself on one side of the equation. This is a crucial step to solve for the unknown variable. Each step is building blocks towards finding the final solution. This process requires attention to detail. So let's distribute!

• Distribute: 5x - 6x - 4 = 15

Next, combine like terms:

• Combine like terms: -x - 4 = 15

Now, let's isolate the x term. Add 4 to both sides:

• Add 4 to both sides: -x = 19

Finally, divide both sides by -1 to solve for x:

• Solve for x: x = -19

Boom! We've found the value of x. It's -19. But wait, we're not done yet, because we still need to find the value of y.

Step 3: Solve for y

We've got x = -19, which is great, but we still need to find the value of y. This is where the first equation comes back into play. Remember, the first equation is already solved for y: y = -3x - 2. All we need to do is plug in the value of x that we just found (-19) into this equation and solve for y. This stage is simple but vital, as it completes the solution to the system. The procedure is like substituting a known value into an equation to find the corresponding value of the other variable. Let's do it step by step, which will help us solve the problem systematically. Substituting the value correctly will allow us to accurately calculate the value of the other variable. It's a fundamental step in solving linear equations and will help us find our final answer. So, here's the substitution:

• Substitute x = -19: y = -3(-19) - 2

Now, let's simplify and solve for y:

• Multiply: y = 57 - 2
• Subtract: y = 55

And there you have it! We've found that y = 55.

Step 4: The Solution

Alright, guys, we're at the finish line! We've calculated the values for both x and y. The solution to the system of equations is the point where the two lines intersect on a graph, and we found it! We've used the substitution method to determine the specific values of x and y that satisfy both equations simultaneously. This means that if we were to graph these two equations, they would intersect at the point (-19, 55). We did it! The solution to the system is an ordered pair, the x-value, and the y-value. It represents the point where the two lines represented by the equations meet on a coordinate plane. These values are the key to unlocking the problem, and they confirm the relationship between the equations. This is where all our work comes together, creating the final answer. Therefore, the solution to the system of equations $\begin{array}{l} y=-3 x-2 \\ 5 x+2 y=15 \end{array}$ is:

• x = -19
• y = 55

Or, as an ordered pair: (-19, 55)

Conclusion

Well done, everyone! We've successfully solved the system of equations using the substitution method. We started with two equations and, through a series of algebraic manipulations, found the values of x and y that satisfied both. This is a powerful demonstration of how we can use mathematical tools to find precise solutions to complex problems. Remember that the substitution method is just one way to solve systems of equations. There are other methods, such as elimination, that you can explore. The key is to practice and become comfortable with different approaches. Practice with various problems is key! With each problem you solve, you'll increase your confidence and develop a deeper understanding of the underlying concepts. So keep practicing, keep exploring, and keep the math adventure going. Don't be afraid to try new problems and challenge yourself. You're doing great!

Now go forth and solve some equations! You got this!