Solving Systems Of Equations Step-by-Step Guide

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Introduction

Hey guys! Let's dive into the exciting world of solving systems of equations. In this article, we're going to break down a specific problem: b. (x = 2y + 7, 3x - 2y = 3). We'll go through the steps in a way that's super easy to understand, even if math isn't your favorite subject. Solving systems of equations is a fundamental skill in algebra, and it's used in all sorts of real-world situations. Think about it: from balancing budgets to designing bridges, the ability to find the values of multiple variables at once is incredibly powerful. So, grab your pencils, and let's get started!

Systems of equations are sets of two or more equations that involve the same variables. The goal is to find the values of these variables that satisfy all equations simultaneously. There are several methods to tackle these problems, but we'll focus on substitution and elimination in this guide. These methods are like tools in your mathematical toolbox, and the more comfortable you are with them, the better equipped you'll be to solve any system of equations that comes your way. The beauty of algebra lies in its structured approach to problem-solving. Each step is logical and builds upon the previous one, leading you to the solution. So, don't be intimidated by the equations; just take it one step at a time, and you'll be amazed at what you can achieve.

Before we jump into the specific problem, let's chat a bit about why this stuff matters. Solving systems of equations isn't just an abstract mathematical exercise. It has practical applications in various fields, including economics, engineering, and computer science. For instance, economists use systems of equations to model supply and demand, engineers use them to analyze circuits, and computer scientists use them in optimization algorithms. By mastering these techniques, you're not just learning math; you're developing skills that can be applied across a wide range of disciplines. Plus, the logical thinking and problem-solving skills you gain will serve you well in any career path you choose. So, whether you're aiming to be an engineer, an economist, or a programmer, understanding systems of equations is a valuable asset. Let's dive in and make sure we've got this down pat!

Understanding the Problem: x = 2y + 7 and 3x - 2y = 3

Okay, let's break down the problem we're tackling: x = 2y + 7 and 3x - 2y = 3. The first equation, x = 2y + 7, tells us that x is equal to two times y plus seven. The second equation, 3x - 2y = 3, gives us another relationship between x and y. Our mission is to find the values of x and y that make both of these equations true at the same time. This means we're looking for a single pair of values (x, y) that works for both equations. Think of it like finding the intersection point of two lines on a graph; that point represents the solution that satisfies both equations.

Before we dive into solving, it's important to appreciate the structure of these equations. The first equation, x = 2y + 7, is already solved for x. This is a huge advantage because it sets us up perfectly for using the substitution method, which we'll discuss in detail shortly. The second equation, 3x - 2y = 3, is in standard form, meaning the variables and constants are neatly arranged. This form is useful for other methods, like elimination, which we'll also explore. Recognizing the form of the equations helps us choose the most efficient method for solving. It's like having the right tool for the job; using a wrench instead of a hammer can make all the difference. So, understanding the structure of the equations is the first step towards finding the solution.

When you see a system of equations like this, don't feel overwhelmed. Remember that each equation is simply a statement about the relationship between the variables. Our job is to decode these statements and find the values that make them both true. It's like solving a puzzle, where each equation is a piece, and the solution is the complete picture. By systematically working through the equations, we can unravel the mystery and find the values of x and y. So, let's roll up our sleeves and start solving! We'll explore the substitution method first, as it's particularly well-suited for this problem, given that one equation is already solved for x. Let's get to it!

Method 1: Solving by Substitution

Alright, let's tackle this system of equations using the substitution method. The substitution method is a fantastic way to solve systems when one equation is already solved for one variable, or when it's easy to isolate a variable. In our case, the first equation, x = 2y + 7, is already solved for x, making substitution a perfect choice. The idea behind substitution is simple: we'll take the expression for x from the first equation and substitute it into the second equation. This will leave us with an equation that only involves y, which we can then solve.

Here's how it works step-by-step:

  1. Substitute: Take the expression for x from the first equation (x = 2y + 7) and plug it into the second equation (3x - 2y = 3). This gives us: 3(2y + 7) - 2y = 3.
  2. Simplify: Now, we need to simplify the equation. Distribute the 3 across the parentheses: 6y + 21 - 2y = 3. Combine like terms: 4y + 21 = 3.
  3. Solve for y: Subtract 21 from both sides: 4y = -18. Divide both sides by 4: y = -18/4, which simplifies to y = -9/2 or -4.5.
  4. Substitute back: Now that we've found the value of y, we can plug it back into either of the original equations to find x. Let's use the first equation, x = 2y + 7: x = 2(-9/2) + 7. Simplify: x = -9 + 7, so x = -2.
  5. Solution: We've found that x = -2 and y = -9/2. So, the solution to the system of equations is the ordered pair (-2, -9/2).

See? The substitution method isn't so scary after all! It's a systematic way to eliminate one variable and solve for the other. By substituting the expression for x into the second equation, we were able to create a single equation with just y, which we easily solved. Then, we plugged the value of y back into one of the original equations to find x. This process demonstrates the power of algebra in breaking down complex problems into simpler steps. Now, let's explore another method to solve this same system, just to show you how versatile these techniques can be. Next up, we'll dive into the elimination method.

Method 2: Solving by Elimination

Alright, let's switch gears and explore another powerful technique for solving systems of equations: the elimination method, sometimes called the addition method. While substitution worked beautifully in this case, elimination is another tool in our mathematical arsenal that can be incredibly useful, especially when equations are in standard form (Ax + By = C). The main idea behind elimination is to manipulate the equations so that when we add them together, one of the variables cancels out, leaving us with a single equation in one variable. Cool, right?

Here’s how we can apply the elimination method to our system:

  1. Align: First, let's rewrite our equations to make sure the x and y terms are aligned:
    • x = 2y + 7 can be rewritten as x - 2y = 7
    • 3x - 2y = 3 (This equation is already in the correct form)
  2. Multiply (if necessary): Notice that the coefficients of y are the same (-2) in both equations. This is great because it means we don't need to multiply either equation by a constant to make the coefficients opposites. If the coefficients weren't the same or opposites, we'd need to multiply one or both equations by a suitable number to make them so.
  3. Eliminate: Since the coefficients of y are the same, we can subtract the second equation from the first to eliminate y. (Alternatively, we could multiply the first equation by -1 and add the equations.) Let's subtract: (x - 2y) - (3x - 2y) = 7 - 3. This simplifies to: x - 2y - 3x + 2y = 4. Combining like terms, we get: -2x = 4.
  4. Solve for x: Divide both sides by -2 to find: x = -2.
  5. Substitute back: Now that we have the value of x, we can substitute it back into either of the original equations to solve for y. Let's use the first original equation, x = 2y + 7: -2 = 2y + 7. Subtract 7 from both sides: -9 = 2y. Divide by 2: y = -9/2 or -4.5.
  6. Solution: We've found that x = -2 and y = -9/2. So, just like with the substitution method, the solution to the system of equations is the ordered pair (-2, -9/2).

See? We arrived at the same solution using a completely different method! This highlights the beauty of mathematics: often, there's more than one way to solve a problem. The elimination method is particularly powerful when the equations are in standard form, and it can sometimes be quicker than substitution, depending on the specific system. The key is to recognize when elimination is a good fit and to carefully manipulate the equations to make a variable cancel out. By mastering both substitution and elimination, you'll be well-equipped to tackle a wide range of systems of equations. Let's wrap things up by summarizing our findings and reinforcing the importance of these skills.

Conclusion and Key Takeaways

Alright, guys, we've successfully solved the system of equations b. (x = 2y + 7, 3x - 2y = 3) using both the substitution and elimination methods! We found that the solution is x = -2 and y = -9/2, or the ordered pair (-2, -4.5). This means that the point (-2, -4.5) is the only pair of values that satisfies both equations simultaneously. High five for that!

Let's recap what we've learned. First, we dove into the substitution method, which is super handy when one equation is already solved for a variable (like x in our case). We substituted the expression for x from the first equation into the second equation, simplified, and solved for y. Then, we plugged the value of y back into one of the original equations to find x. It's like a mathematical chain reaction, where one substitution leads to another until we crack the code.

Next, we explored the elimination method, which shines when equations are in standard form (Ax + By = C). We aligned the equations, made sure the coefficients of one variable were the same or opposites, and then either added or subtracted the equations to eliminate that variable. This left us with a single equation in one variable, which we solved easily. Then, just like with substitution, we plugged the value we found back into one of the original equations to find the other variable.

The key takeaway here is that understanding multiple methods gives you flexibility and empowers you to choose the most efficient approach for a given problem. There's no one-size-fits-all solution in math, and knowing different techniques is like having a well-stocked toolbox. Plus, practicing both substitution and elimination helps you develop a deeper understanding of how systems of equations work and strengthens your overall problem-solving skills.

Remember, solving systems of equations isn't just about getting the right answer; it's about developing logical thinking and analytical skills that will benefit you in countless areas of life. Whether you're balancing a budget, planning a project, or just trying to figure out the best route to take during rush hour, the ability to analyze relationships between variables and find solutions is a valuable asset. So, keep practicing, keep exploring, and keep challenging yourself. You've got this!