Solving Systems Of Equations A Step-by-Step Guide

Hey guys! Ever get stuck trying to solve a system of equations? It can feel like you're trying to crack a secret code, right? But don't worry, we're here to break it down and make it super easy. In this article, we'll tackle a common type of problem: solving a system of equations. We'll walk through a specific example step-by-step, so you'll not only understand the how but also the why behind each move. So, let's dive in and unlock the secrets of solving equations!

Understanding Systems of Equations

Before we jump into solving, let's quickly recap what a system of equations actually is. Think of it as a set of two or more equations that share the same variables. Our goal is to find the values for those variables that make all the equations true at the same time. It's like finding the perfect combo that unlocks every door!

In our case, we have two equations:

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

We need to find the x and y values that satisfy both of these equations simultaneously. There are a few methods to do this, but we'll focus on the substitution method here because it's super handy for this particular problem. Why? Because the first equation already tells us what y is equal to!

The substitution method is like a clever shortcut. We're going to use the information from one equation to rewrite the other equation, making it easier to solve. It's like using a secret ingredient to transform a recipe. The key is to identify an equation where one variable is already isolated (like our y in the first equation) and then substitute that expression into the other equation.

Why does this work? Well, if y truly equals -3x - 2, then we can replace y with that entire expression in the second equation without changing the solution. It's like swapping out identical puzzle pieces – the overall picture remains the same. This substitution will leave us with an equation that only involves x, which we can then solve using basic algebra. Once we have x, we can plug it back into either of the original equations to find y. Easy peasy!

Step-by-Step Solution Using Substitution

Let's break down the solution step-by-step so you can follow along and see exactly how it works.

Step 1: Substitute the Expression for y

The first equation, y = -3x - 2, is our golden ticket. It tells us exactly what y is in terms of x. So, we're going to take this expression (-3x - 2) and substitute it for y in the second equation:

5x + 2(y) = 15

Becomes:

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

See what we did there? We replaced the y with its equivalent expression. This is the heart of the substitution method. By doing this, we've transformed our system of two equations into a single equation with just one variable (x). Now, we're on familiar territory – solving a regular equation!

It's super important to use parentheses when you substitute, especially when dealing with expressions that have more than one term. The parentheses ensure that we distribute the multiplication correctly in the next step. Think of it like building a fence around the expression we're substituting – we want to keep it all together and treat it as a single unit.

Step 2: Simplify and Solve for x

Now that we've substituted, it's time to simplify the equation and solve for x. This involves a little bit of algebraic maneuvering, but don't worry, we'll take it slow.

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

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

Becomes:

5x - 6x - 4 = 15

Remember the distributive property? It's like sharing the love (or the multiplication, in this case) with everyone inside the parentheses. We multiply the 2 by both the -3x and the -2.

Next, we combine like terms on the left side of the equation:

5x - 6x - 4 = 15

Becomes:

-x - 4 = 15

We have 5 x's and we're taking away 6 x's, leaving us with -1 x, which we simply write as -x.

Now, we want to isolate x on one side of the equation. To do this, we add 4 to both sides:

-x - 4 = 15

Becomes:

-x = 19

Almost there! We have -x = 19, but we want x by itself. To get rid of the negative sign, we multiply both sides of the equation by -1:

-x = 19

Becomes:

x = -19

Woohoo! We've found the value of x. It's like finding one piece of the puzzle. But we're not done yet – we still need to find y.

Step 3: Substitute the Value of x to Find y

Now that we know x = -19, we can plug this value back into either of the original equations to solve for y. The first equation, y = -3x - 2, looks simpler, so let's use that one:

y = -3x - 2

Substitute x = -19:

y = -3(-19) - 2

Now, we just need to simplify:

y = 57 - 2

y = 55

Fantastic! We've found y as well. It's like finding the last piece of the puzzle, and everything is starting to come together.

Step 4: Write the Solution as an Ordered Pair

We've found x = -19 and y = 55. Now, we need to express our solution in the standard form for a system of equations: as an ordered pair (x, y).

So, our solution is (-19, 55).

This ordered pair represents the point where the two lines represented by our equations intersect on a graph. It's the one and only point that satisfies both equations simultaneously. Think of it like the secret meeting place where the two lines agree!

Verifying the Solution

Before we celebrate our victory, it's always a good idea to verify our solution. This means plugging our values for x and y back into the original equations to make sure they hold true. It's like double-checking our work to catch any sneaky mistakes.

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

Great! 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

Awesome! The second equation also holds true. Since our solution satisfies both original equations, we can be confident that we've found the correct answer.

Common Mistakes to Avoid

Solving systems of equations can be tricky, and there are a few common pitfalls that students often encounter. But don't worry, we're here to help you avoid them!

• Forgetting to Distribute: When substituting an expression into an equation, remember to distribute any multiplication across all terms inside the parentheses. This is a classic mistake, so double-check that you've multiplied correctly.
• Incorrectly Combining Like Terms: Be careful when combining like terms, especially when dealing with negative signs. Make sure you're adding or subtracting the coefficients correctly. It's like counting apples and oranges – you need to keep track of what you're adding together.
• Solving for x but Forgetting to Solve for y: Remember, the goal is to find both x and y. Don't stop after you've found one variable – plug it back into an equation to find the other one. It's like finding one half of a treasure map – you still need the other half to locate the treasure!
• Not Verifying the Solution: Always, always, always verify your solution by plugging it back into the original equations. This is the best way to catch any errors and ensure that you have the correct answer. It's like proofreading your writing before you submit it – a quick check can make a big difference.

Practice Makes Perfect

Solving systems of equations is a skill that gets better with practice. The more you work through problems, the more comfortable you'll become with the different methods and techniques. So, don't be afraid to tackle lots of examples!

Try solving other systems of equations using the substitution method. You can also explore other methods, like elimination, to see which one works best for you in different situations. It's like learning different dance moves – the more you know, the more versatile you'll be on the dance floor (or in the math classroom!).

Conclusion

So, there you have it! We've walked through a step-by-step solution to a system of equations using the substitution method. Remember, the key is to substitute one expression for a variable in the other equation, simplify, solve for the remaining variable, and then plug that value back in to find the other variable. And don't forget to verify your solution!

Solving systems of equations might seem daunting at first, but with practice and a clear understanding of the steps involved, you'll be solving them like a pro in no time. Keep practicing, and remember, math can be fun! You got this! If you have any questions please don't hesitate to ask.