Solving System Of Equations: A Step-by-Step Guide
Hey guys! Today, we're diving into the fascinating world of solving systems of equations. Specifically, we're going to tackle a system with two equations: y = 5x + 24 and y = 9x. Don't worry if this looks intimidating – we'll break it down into easy-to-follow steps. Think of it as a puzzle where we need to find the values of 'x' and 'y' that make both equations true at the same time. So, grab your pencils, and let's get started!
Understanding Systems of Equations
Before we jump into solving, let's quickly understand what a system of equations actually is. A system of equations is simply a set of two or more equations that share the same variables. In our case, we have two equations, and the variables are 'x' and 'y'. The solution to a system of equations is the set of values for the variables that satisfy all equations in the system simultaneously. Think of it like finding the intersection point of two lines on a graph – that point represents the (x, y) values that work for both equations.
There are several methods to solve systems of equations, including substitution, elimination, and graphing. We'll be focusing on the substitution method in this article because it's particularly well-suited for the system we're working with. This method involves solving one equation for one variable and then substituting that expression into the other equation. This effectively reduces the problem to a single equation with a single variable, which is much easier to solve.
When dealing with systems of equations, remember that the goal is to find the specific values for each variable that make all the equations in the system true at the same time. This concept is fundamental in various fields, including mathematics, physics, engineering, and economics, where real-world problems often involve multiple interconnected relationships. Mastering the techniques for solving systems of equations, like substitution, is therefore a valuable skill for anyone pursuing studies or careers in these areas. The beauty of the substitution method lies in its systematic approach, allowing us to break down complex problems into smaller, more manageable steps. By substituting one variable's expression into another equation, we effectively eliminate one variable, making it easier to isolate and solve for the remaining one. This process is not only efficient but also provides a clear and logical path to the solution, reinforcing our understanding of the underlying mathematical principles.
The Substitution Method: Step-by-Step
Okay, let's get our hands dirty and solve the system using the substitution method. This method is super handy when one of the equations is already solved for a variable (like our 'y' in this case!). Here's how we'll tackle it, step-by-step:
Step 1: Identify the Equations
First, let's clearly write down our equations:
- Equation 1: y = 5x + 24
- Equation 2: y = 9x
See how both equations are already solved for 'y'? This makes the substitution method a perfect choice for this problem. We've got a head start already, guys!
Step 2: Substitute
Since both equations are equal to 'y', we can set them equal to each other. This is the heart of the substitution method. We're essentially saying that if y is the same in both equations, then the right-hand sides of the equations must also be equal. So, we have:
5x + 24 = 9x
Boom! We've now got a single equation with only one variable ('x'). This is a huge step forward because we can now solve for 'x' without worrying about 'y' just yet. By substituting one expression for another, we've effectively simplified the problem into a more manageable form. This step highlights the power of the substitution method in reducing the complexity of a system of equations. It's like taking a tangled mess of string and carefully untangling it, one strand at a time, until you have a clear, straight line. The key is to recognize the relationship between the equations and use that relationship to your advantage, making the solution process much smoother and more efficient.
Step 3: Solve for 'x'
Now, let's solve the equation 5x + 24 = 9x for 'x'. This is just a simple algebraic equation, so we'll use our trusty algebra skills.
First, we want to get all the 'x' terms on one side of the equation. Let's subtract 5x from both sides:
5x + 24 - 5x = 9x - 5x
This simplifies to:
24 = 4x
Next, to isolate 'x', we'll divide both sides by 4:
24 / 4 = 4x / 4
This gives us:
x = 6
Huzzah! We've found the value of 'x'. It's like finding a key piece of the puzzle. Now we're one step closer to unlocking the complete solution. The process of solving for 'x' here demonstrates the importance of algebraic manipulation in simplifying equations. By performing operations like subtracting and dividing on both sides, we maintain the equality of the equation while isolating the variable we want to solve for. This step-by-step approach ensures that we can confidently arrive at the correct value for 'x', which is crucial for the next part of our solution. This careful manipulation is the cornerstone of solving equations, allowing us to transform a complex equation into a simple, easily understood form.
Step 4: Substitute 'x' to Find 'y'
Now that we know x = 6, we can plug this value back into either Equation 1 or Equation 2 to find the value of 'y'. Let's use Equation 2 because it looks a bit simpler:
y = 9x
Substitute x = 6:
y = 9 * 6
y = 54
Awesome! We've found 'y' too. We're on a roll, guys! Plugging the value of 'x' back into one of the original equations is a crucial step in the substitution method. It allows us to leverage the value we've already found to determine the value of the other variable. By choosing the simpler equation, like Equation 2 in this case, we can often make the calculation easier and reduce the chance of errors. This step highlights the interconnectedness of the variables in a system of equations. Once we know the value of one variable, we can use it as a stepping stone to find the value of the other, completing the puzzle and arriving at the final solution. This process not only provides the answer but also reinforces the understanding of how the equations in the system relate to each other.
Step 5: Check Your Solution
It's always a good idea to check our solution to make sure we haven't made any silly mistakes. To do this, we'll plug both x = 6 and y = 54 into both equations and see if they hold true.
- Equation 1: y = 5x + 24
- 54 = 5(6) + 24
- 54 = 30 + 24
- 54 = 54 (Yep, it checks out!)
- Equation 2: y = 9x
- 54 = 9(6)
- 54 = 54 (Perfect!)
Both equations are true with our values for 'x' and 'y'. We did it! Checking our solution is a vital step in any mathematical problem-solving process. It acts as a safety net, ensuring that we haven't made any calculation errors along the way. By plugging the values we found back into the original equations, we can verify that they satisfy all the conditions of the system. This step not only confirms the correctness of our answer but also deepens our understanding of the problem and the solution process. It's like double-checking your work before submitting it – a little extra effort can make a big difference in ensuring accuracy. This practice instills confidence in our solution and reinforces the importance of precision in mathematical calculations.
The Solution
So, the solution to the system of equations is x = 6 and y = 54. We can write this as an ordered pair (6, 54). This is 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.
Visualizing the Solution (Optional)
If you're a visual learner, it can be helpful to think about what this solution means graphically. Each equation in our system represents a straight line. The solution (6, 54) is the point where these two lines cross each other. You could actually graph these lines on a coordinate plane to see this intersection visually. This provides a deeper understanding of the concept and connects the algebraic solution with a geometric representation.
Conclusion
Solving systems of equations might seem tricky at first, but with the substitution method (and a little practice!), it becomes a piece of cake. We successfully found the values of 'x' and 'y' that make both equations true. Remember, the key is to break down the problem into smaller steps, substitute carefully, and always check your solution. You guys got this!
Keep practicing, and you'll become a system-solving pro in no time! And remember, math can be fun when you approach it with a positive attitude and a willingness to learn. So, keep exploring, keep questioning, and keep solving! You're all capable of amazing things, and mastering skills like solving systems of equations is just one step on your journey to mathematical excellence. The more you practice, the more confident you'll become, and the more you'll appreciate the power and beauty of mathematics. So, don't be afraid to tackle new challenges – each problem you solve is a victory that builds your skills and your confidence. Keep up the great work, and remember, the world of mathematics is full of exciting discoveries waiting to be made!
