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

Nov 23, 2025 by Andrew McMorgan 51 views

Hey guys! Ever find yourselves staring blankly at a system of equations, wondering where to even begin? Don't worry, you're not alone! Systems of equations can seem intimidating at first, but with a little know-how, they're totally solvable. In this article, we're going to break down a specific system of equations step-by-step, so you can conquer similar problems with confidence. Let's dive in!

Understanding the System of Equations

Before we jump into solving, let's take a closer look at the system we're dealing with. We have two equations, each with two variables, x and y:

y = -3x - 2
5x + 2y = 15

A system of equations like this represents two or more relationships between the variables. Our goal is to find the values of x and y that satisfy both equations simultaneously. Think of it as finding the point where the lines represented by these equations intersect on a graph. There are several methods we can use to solve systems of equations, including substitution, elimination, and graphing. For this particular system, we're going to use the substitution method, which is often the most efficient approach when one of the equations is already solved for one variable (like our first equation, which is solved for y). The beauty of the substitution method lies in its simplicity: we'll essentially replace a variable in one equation with its equivalent expression from the other equation. This will leave us with a single equation with a single variable, which we can easily solve. Once we find the value of that variable, we can plug it back into either of the original equations to find the value of the other variable. This step-by-step approach makes even complex systems of equations manageable, and with practice, you'll be solving them like a pro in no time!

Step 1: Substitution – Plugging in the Value of y

The substitution method is our weapon of choice here, and it's perfect because our first equation, y = -3x - 2, already gives us y in terms of x. This is a huge advantage! We can take this expression for y and substitute it directly into the second equation, 5x + 2y = 15. This is where the magic happens. By replacing y in the second equation, we eliminate one variable and create a new equation with only x. This new equation will be much easier to solve. Think of it like this: we're taking the information we have about the relationship between x and y from the first equation and using it to simplify the second equation. The substitution process looks like this:

5x + 2(-3x - 2) = 15

Notice how we've replaced the y in the second equation with the entire expression (-3x - 2). It's crucial to put this expression in parentheses to ensure we distribute the 2 correctly in the next step. This substitution is the key to unlocking the solution. It transforms a two-variable problem into a single-variable problem, which we can then tackle using basic algebraic techniques. This step highlights the power of substitution – by strategically replacing variables, we can simplify complex equations and pave the way for a straightforward solution. Mastering this technique will significantly boost your equation-solving skills!

Step 2: Simplifying and Solving for x

Now that we've substituted, it's time to simplify the equation and isolate x. Remember our equation from the last step?

5x + 2(-3x - 2) = 15

First, we need to distribute the 2 across the parentheses:

5x - 6x - 4 = 15

Next, combine the x terms:

-x - 4 = 15

Now, let's isolate the x term by adding 4 to both sides of the equation:

-x = 19

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

x = -19

And there you have it! We've successfully solved for x. This step demonstrates the importance of following the order of operations and using algebraic manipulations to isolate the variable we're trying to find. Distributing, combining like terms, and performing inverse operations are all essential tools in our equation-solving arsenal. By carefully applying these techniques, we've transformed a seemingly complex equation into a simple solution. This ability to simplify and solve for a variable is a fundamental skill in algebra and will serve you well in countless mathematical problems. So, embrace the process, practice these steps, and watch your equation-solving confidence soar!

Step 3: Finding y Using the Value of x

We've nailed down the value of x, which is -19. Awesome! But remember, our ultimate goal is to find the values of both x and y that satisfy the system of equations. So, what's the next step? It's simple: we'll use the value of x we just found and plug it back into one of the original equations to solve for y. The beauty of this step is that you can choose either equation – they'll both lead you to the correct value of y. However, to make things as easy as possible, it's often best to choose the equation that's already solved for y, which in our case is:

y = -3x - 2

This equation is perfectly set up for us to substitute the value of x and calculate y. Let's do it!

y = -3(-19) - 2

First, multiply -3 by -19:

y = 57 - 2

Then, subtract 2 from 57:

y = 55

Boom! We've found the value of y: it's 55. This step highlights the interconnectedness of the variables in a system of equations. Once we solve for one variable, we can use that information to unlock the value of the other. This back-substitution technique is a powerful tool that allows us to complete the solution process and find the complete picture.

Step 4: The Solution and Verification

We've done it! We've successfully solved the system of equations. Our solution is:

x = -19 y = 55

This means that the point (-19, 55) is the solution to our system. It's the point where the lines represented by our two equations intersect. But before we celebrate too much, there's one crucial step we need to take: verification. It's always a good idea to check our solution to make sure it's correct. This helps prevent errors and gives us confidence in our answer. To verify our solution, we simply plug the values of x and y we found back into both of the original equations. If both equations hold true, then our solution is correct.

Let's start with the first equation:

y = -3x - 2

Substitute x = -19 and y = 55:

55 = -3(-19) - 2

Simplify:

55 = 57 - 2

55 = 55

The first equation checks out! Now, let's try the second equation:

5x + 2y = 15

Substitute x = -19 and y = 55:

5(-19) + 2(55) = 15

Simplify:

-95 + 110 = 15

15 = 15

The second equation also checks out! Since our solution satisfies both equations, we can confidently say that (-19, 55) is the correct solution to the system of equations. This verification step is a testament to the importance of accuracy and attention to detail in mathematics. By taking the time to check our work, we can ensure that our solutions are not only correct but also meaningful in the context of the problem.

Conclusion: Mastering Systems of Equations

Solving systems of equations might seem tricky at first, but as we've seen, breaking it down into manageable steps makes it totally achievable. We tackled the system:

y = -3x - 2
5x + 2y = 15

Using the substitution method, we found that x = -19 and y = 55. Remember, the key is to substitute, simplify, solve, and verify! Understanding these steps empowers you to solve a wide range of problems. Keep practicing, and you'll become a system-of-equations superstar in no time. So, go forth and conquer those equations, guys! You've got this!