Solve Equations: Step-by-Step Guide

by Henrik Larsen 36 views

Hey guys! Today, we're diving into the world of algebra to tackle a common problem: solving a system of equations. If you've ever felt lost in a maze of x's and y's, don't worry – we're going to break it down step by step. We'll use the substitution method to make things super clear and easy to understand. So, grab your pencils and let's get started!

Understanding Systems of Equations

Before we jump into the solution, let's quickly recap what a system of equations actually is. Simply put, a system of equations is a set of two or more equations containing the same variables. Our goal is to find values for these variables that make all equations in the system true simultaneously. Think of it like finding the perfect combination that unlocks all the equations at once. In our case, we're dealing with two equations and two variables (x and y), which is a pretty common scenario.

Why Solve Systems of Equations?

You might be wondering, "Why bother solving these things?" Well, systems of equations pop up in all sorts of real-world situations! They're used in everything from calculating the optimal mix of ingredients in a recipe to modeling complex scientific phenomena. Understanding how to solve them is a valuable skill that opens doors to many fields. For instance, in economics, you might use systems of equations to determine market equilibrium, where supply and demand intersect. In engineering, they can help calculate forces and stresses in structures. Even in computer graphics, systems of equations play a role in creating realistic images and animations.

The beauty of mathematics lies in its ability to model real-world problems, and systems of equations are a prime example of this. They allow us to express relationships between different quantities and find specific solutions that satisfy multiple conditions. So, as you learn to solve these equations, remember that you're not just manipulating abstract symbols; you're gaining a powerful tool for understanding and solving problems in the world around you. The ability to translate a real-world problem into a mathematical model and then solve it is a crucial skill in many professions, and mastering systems of equations is a significant step in that direction. So, let's dive in and see how we can tackle the system of equations at hand!

The Equations We're Tackling

Let's take a look at the specific system of equations we'll be working with today:

4x + 2 = -2y
6y - 18 = 6x

These two equations represent two lines on a graph, and the solution to the system is the point where these lines intersect. Our job is to find the x and y coordinates of that intersection point algebraically, without having to draw a graph. There are several methods to solve systems of equations, but we'll be focusing on the substitution method in this guide. This method is particularly useful when one of the equations can be easily rearranged to solve for one variable in terms of the other.

Why the Substitution Method?

The substitution method is a fantastic tool in our equation-solving arsenal because it's straightforward and versatile. It's especially handy when one of the equations has a variable that's already isolated or can be easily isolated. In such cases, substitution can simplify the process significantly. The basic idea behind the substitution method is to solve one equation for one variable, and then substitute that expression into the other equation. This eliminates one variable, leaving us with a single equation in a single variable, which we can then solve. Once we've found the value of that variable, we can plug it back into either of the original equations to find the value of the other variable. It's like a domino effect – solve for one, and the other falls into place!

Another reason to love the substitution method is its logical flow. It breaks down the problem into smaller, manageable steps. This makes it easier to track your progress and avoid errors. It's also a great method for understanding the underlying relationship between the variables in the system. By expressing one variable in terms of the other, we gain insight into how they interact and influence each other. This can be particularly useful in real-world applications where understanding the relationships between variables is just as important as finding the numerical solutions.

Step 1: Solve One Equation for One Variable

The first step in using the substitution method is to choose one of the equations and solve it for one of the variables. Looking at our system:

4x + 2 = -2y
6y - 18 = 6x

The first equation, 4x + 2 = -2y, seems like a good candidate because we can easily isolate y. Let's do that:

  1. Start with the equation: 4x + 2 = -2y
  2. Divide both sides by -2: (4x + 2) / -2 = (-2y) / -2
  3. Simplify: -2x - 1 = y

So, we've now solved the first equation for y: y = -2x - 1. This is a crucial step because we've expressed y in terms of x, which means we can substitute this expression into the other equation.

Choosing the Right Equation and Variable

While we chose the first equation to solve for y, it's important to realize that there are often multiple ways to approach this step. We could have also chosen the second equation, 6y - 18 = 6x, and solved for either x or y. The key is to look for the option that will lead to the simplest algebraic manipulation. For example, if one equation has a variable with a coefficient of 1, solving for that variable is usually a good idea because it avoids fractions. Similarly, if one equation already has a variable mostly isolated, that's often a sign that it's the right choice.

In our case, solving the first equation for y was a relatively straightforward process. However, let's briefly consider what would have happened if we had chosen to solve the second equation for x. We would have started with 6y - 18 = 6x and then divided both sides by 6 to get y - 3 = x. This is also a perfectly valid approach, and we would eventually arrive at the same solution. The choice often comes down to personal preference and what feels most comfortable to you. The important thing is to understand the flexibility you have in choosing which equation and variable to solve for, and to pick the option that seems most efficient.

Step 2: Substitute into the Other Equation

Now that we have y = -2x - 1, we can substitute this expression for y into the other equation (the one we didn't use in Step 1). This is the heart of the substitution method. Remember, we're trying to eliminate one variable so we can solve for the other.

Our second equation is 6y - 18 = 6x. Let's substitute -2x - 1 for y:

6(-2x - 1) - 18 = 6x

Notice how we've replaced y with its equivalent expression in terms of x. This gives us a new equation that only contains the variable x, which we can solve.

The Importance of the "Other" Equation

It's crucial to substitute the expression into the other equation, not the one you just used to solve for the variable. If you substitute back into the same equation, you'll end up with a tautology (something that's always true, like 0 = 0), which doesn't help you solve for the variables. The whole point of substitution is to combine the information from both equations, and you can only do that by substituting into the equation you haven't used yet.

Think of it like this: you've taken a piece of information from one equation (the relationship between x and y in the first equation) and plugged it into the second equation. This allows you to see how that relationship affects the other equation and ultimately find the specific values of x and y that satisfy both equations simultaneously. So, always remember to substitute into the other equation to keep the process moving forward.

Step 3: Solve for the Remaining Variable

We now have the equation 6(-2x - 1) - 18 = 6x. Let's solve for x:

  1. Distribute the 6: -12x - 6 - 18 = 6x
  2. Combine like terms: -12x - 24 = 6x
  3. Add 12x to both sides: -24 = 18x
  4. Divide both sides by 18: x = -24 / 18
  5. Simplify: x = -4 / 3

So, we've found the value of x: x = -4/3. This is a major step forward! We're halfway to the solution. Now we just need to find the corresponding value of y.

The Power of Simplification

Notice how we simplified the equation at each step. This is a key strategy in algebra. Simplifying makes the equation easier to work with and reduces the chances of making errors. Combining like terms, distributing, and reducing fractions are all important simplification techniques. In this case, simplifying the fraction -24/18 to -4/3 not only makes the answer look cleaner but also makes it easier to use in the next step.

Algebra is often about taking a complex expression and breaking it down into its simplest form. This not only makes it easier to solve but also provides a deeper understanding of the relationships between the variables. So, always be on the lookout for opportunities to simplify your equations. It's a skill that will serve you well in all areas of mathematics.

Step 4: Substitute Back to Find the Other Variable

We know that x = -4/3. Now we can substitute this value back into either of the original equations (or the equation we solved for y in Step 1) to find the value of y. Let's use the equation y = -2x - 1 because it's already solved for y:

  1. Substitute x = -4/3: y = -2(-4/3) - 1
  2. Multiply: y = 8/3 - 1
  3. Find a common denominator: y = 8/3 - 3/3
  4. Subtract: y = 5/3

So, we've found the value of y: y = 5/3.

Choosing the Right Equation for Back-Substitution

As mentioned earlier, you can substitute the value of x back into any of the equations to find y. However, some equations might be easier to work with than others. In this case, using the equation y = -2x - 1 was the most straightforward because it was already solved for y. This avoided the need for any additional algebraic manipulation. If we had chosen to use one of the original equations, we would have had to do a bit more work to isolate y after substituting the value of x.

The key is to look for the equation that will minimize the number of steps required to solve for the remaining variable. This can save you time and reduce the risk of making errors. With practice, you'll develop a sense for which equations are the most convenient to use in different situations. But remember, regardless of which equation you choose, you should arrive at the same answer for y.

Step 5: Check Your Solution

It's always a good idea to check your solution to make sure it's correct. To do this, substitute the values of x and y into both of the original equations. If both equations are true, then your solution is correct.

Let's check our solution x = -4/3 and y = 5/3:

  • Equation 1: 4x + 2 = -2y
    • Substitute: 4(-4/3) + 2 = -2(5/3)
    • Simplify: -16/3 + 6/3 = -10/3
    • -10/3 = -10/3 (True!)
  • Equation 2: 6y - 18 = 6x
    • Substitute: 6(5/3) - 18 = 6(-4/3)
    • Simplify: 10 - 18 = -8
    • -8 = -8 (True!)

Since our solution satisfies both equations, we know it's correct.

The Importance of Verification

Checking your solution is a crucial step in the problem-solving process. It's like the final quality control check that ensures you haven't made any mistakes along the way. Even if you're confident in your calculations, it's always worth taking a few minutes to verify your answer. This is especially important in exams or assignments where accuracy is paramount.

By substituting your solution back into the original equations, you're essentially reversing the process and seeing if everything lines up. If your solution doesn't satisfy both equations, it means there's an error somewhere in your calculations, and you need to go back and find it. This could be a simple arithmetic mistake, a sign error, or a more fundamental misunderstanding of the problem. Whatever the cause, catching it early can save you from losing points or making incorrect decisions based on faulty results.

The Solution

The solution to the system of equations is x = -4/3 and y = 5/3. We can write this as an ordered pair: (-4/3, 5/3). This is the point where the two lines represented by the equations intersect on a graph.

Visualizing the Solution

It's helpful to visualize what this solution means graphically. Each of our original equations represents a straight line. The solution to the system of equations is the point where these two lines intersect. So, the ordered pair (-4/3, 5/3) represents the coordinates of that intersection point. If you were to graph these two lines, you would see that they cross each other at exactly this point.

This graphical interpretation can provide a deeper understanding of systems of equations. It shows that solving a system of equations is essentially finding the common ground between two or more relationships. The intersection point represents the values of the variables that satisfy all the equations simultaneously. This visual representation can also be helpful in solving real-world problems, where the lines might represent constraints or conditions, and the intersection point represents the optimal solution that meets all the requirements.

Conclusion

We've successfully solved the system of equations using the substitution method! Remember the key steps:

  1. Solve one equation for one variable.
  2. Substitute into the other equation.
  3. Solve for the remaining variable.
  4. Substitute back to find the other variable.
  5. Check your solution.

With practice, you'll become a pro at solving systems of equations. Keep practicing, and you'll be tackling even more complex problems in no time! Solving systems of equations is a fundamental skill in algebra and has applications in various fields. By mastering this skill, you're not just learning a mathematical technique; you're developing a problem-solving mindset that will serve you well in many aspects of life.

So, keep practicing, keep exploring, and don't be afraid to ask questions. The world of mathematics is full of fascinating challenges and rewarding discoveries, and you're well on your way to becoming a confident and capable mathematician. Keep up the great work!