Solving Systems Of Linear Equations 4x - Y = 13 And 2x - 3y = 19 With The Combined Method
Hey guys! Math can sometimes feel like a puzzle, but don't worry, we'll crack this one together! Today, we're diving into solving a system of two-variable linear equations using the combined method. Think of it as having a super-tool that mixes and matches different techniques to get to the solution. Let's jump right in!
Understanding the Problem: What are we dealing with?
Before we start crunching numbers, let's understand what we're looking at. We have a system of two linear equations:
- 4x - y = 13
- 2x - 3y = 19
What does this mean? Well, each equation represents a straight line on a graph. The solution to this system of equations is the point (or points) where these lines intersect. In other words, we're looking for the values of x and y that satisfy both equations simultaneously.
Why use the combined method? Great question! The combined method is awesome because it leverages the strengths of both the substitution and elimination methods. Sometimes, one method might be easier to apply than the other depending on the specific equations. The combined method gives us the flexibility to choose the best approach for each situation. It's like having a Swiss Army knife for solving equations!
Think about it this way: substitution is great when one of the variables is already isolated or easy to isolate. Elimination, on the other hand, shines when the coefficients of one of the variables are the same or easy to make the same. The combined method lets us switch between these approaches as needed, making the process more efficient and less prone to errors.
Breaking Down Linear Equations
Let's take a moment to really understand what these linear equations are telling us. The equation 4x - y = 13 is saying that if you take four times the value of x and subtract the value of y, you'll get 13. Similarly, 2x - 3y = 19 tells us that two times x minus three times y equals 19.
Each of these equations represents a line on a graph. The solutions to the equation are all the points (x, y) that lie on that line. When we have a system of two equations, we're essentially looking for the point where the two lines cross. This point of intersection is the solution that satisfies both equations.
Why is this important? Well, systems of linear equations pop up everywhere in the real world! From figuring out the cost of different items when you have a budget to planning the logistics of a project, understanding how to solve these equations is a super valuable skill.
Preparing for the Combined Method
Before we dive into the combined method, let's make sure we're all on the same page with the basics. We need to be comfortable with manipulating equations – adding, subtracting, multiplying, and dividing both sides – without changing their fundamental meaning. Remember, the golden rule is that whatever you do to one side of the equation, you must do to the other!
We also need to be familiar with the substitution and elimination methods individually. While we're using a combined approach, understanding the underlying principles of these methods will help us make informed decisions about which technique to use at each step. Think of it as having a solid foundation before building a house – it makes the whole process smoother and more stable.
So, with our understanding of linear equations and the strategic advantage of the combined method, let's get ready to tackle the problem step-by-step!
Step-by-Step Solution using the Combined Method
Okay, let's get our hands dirty and solve this system of equations using the combined method. We'll break it down into easy-to-follow steps.
Step 1: Choose a Method for the First Variable
Looking at our equations:
- 4x - y = 13
- 2x - 3y = 19
We can see that the coefficient of y in the first equation is -1, which makes it relatively easy to isolate y. This suggests that the substitution method might be a good starting point. Alternatively, we could consider the elimination method by multiplying the second equation by 2 to make the coefficients of x match.
For this example, let's start with the elimination method. It's a matter of preference, and sometimes one method will be more straightforward than the other depending on the problem. The key is to recognize the options and choose the path that seems most efficient for you.
Why elimination first? In this case, multiplying the second equation by a constant will easily make the x coefficients match. This allows us to eliminate x and solve for y. It's a strategic choice that can simplify the process.
Step 2: Elimination of the First Variable (x)
To eliminate x, we'll multiply the second equation by -2:
-2 * (2x - 3y) = -2 * 19
This gives us:
-4x + 6y = -38
Now we have a modified second equation. Let's rewrite our system:
- 4x - y = 13
- -4x + 6y = -38
Notice that the coefficients of x are now opposites (4 and -4). This is perfect for elimination! We can add the two equations together:
(4x - y) + (-4x + 6y) = 13 + (-38)
This simplifies to:
5y = -25
Step 3: Solve for the Remaining Variable (y)
Now we have a simple equation with just one variable. We can easily solve for y by dividing both sides by 5:
y = -25 / 5
y = -5
Great! We've found the value of y. This is a major step forward in solving our system.
Step 4: Substitute to Find the Other Variable (x)
Now that we know y = -5, we can substitute this value into either of the original equations to solve for x. Let's use the first equation:
4x - y = 13
Substitute y = -5:
4x - (-5) = 13
Simplify:
4x + 5 = 13
Subtract 5 from both sides:
4x = 8
Divide by 4:
x = 2
Fantastic! We've found the value of x. We now have both x and y.
Step 5: Write the Solution as an Ordered Pair
The solution to the system of equations is the ordered pair (x, y). In our case, we found x = 2 and y = -5. So, the solution is:
(2, -5)
This means that the point (2, -5) is the intersection of the two lines represented by our original equations. It's the one and only point that satisfies both equations simultaneously.
Verification: Checking our Solution
Okay, we've got our solution, but let's be absolutely sure it's correct. The best way to do this is to plug our values of x and y back into the original equations and see if they hold true. This step is crucial to avoid errors and ensure we have the correct answer.
Plugging into the First Equation
Let's start with the first equation:
4x - y = 13
Substitute x = 2 and y = -5:
4(2) - (-5) = 13
Simplify:
8 + 5 = 13
13 = 13
Awesome! The equation holds true. This is a good sign that our solution is on the right track.
Plugging into the Second Equation
Now let's check the second equation:
2x - 3y = 19
Substitute x = 2 and y = -5:
2(2) - 3(-5) = 19
Simplify:
4 + 15 = 19
19 = 19
Excellent! The second equation also holds true. This confirms that our solution (2, -5) is indeed correct.
Why is verification so important? Think of it as double-checking your work on a test. It's easy to make a small mistake during the solving process, especially with multiple steps. Verification catches these errors and gives us confidence in our answer. It's a best practice that every math student should adopt.
The Importance of Precision
In mathematics, precision is key. A small error can throw off the entire solution. That's why it's crucial to be careful with each step and double-check your work. When dealing with negative signs, fractions, or any other potentially tricky elements, take extra care.
Verification is not just about confirming your answer; it's also about developing good mathematical habits. By consistently checking your solutions, you'll become more accurate and confident in your problem-solving abilities.
So, remember, always verify your solutions! It's the final piece of the puzzle that ensures you've got the right answer.
Conclusion: Mastering the Combined Method
Alright, guys, we did it! We successfully solved the system of equations using the combined method. We've seen how powerful this approach can be, leveraging both substitution and elimination techniques to find the solution. Remember, the key is to be flexible and choose the method that best suits the problem at hand.
Recap of the Steps
Let's quickly recap the steps we took:
- Choose a Method: We started by assessing the equations and deciding whether elimination or substitution would be the most efficient first step. In our case, we opted for elimination.
- Eliminate a Variable: We manipulated one or both equations to make the coefficients of one variable opposites, then added the equations to eliminate that variable.
- Solve for the Remaining Variable: We solved the resulting equation for the remaining variable.
- Substitute: We substituted the value we found back into one of the original equations to solve for the other variable.
- Write the Solution: We wrote the solution as an ordered pair (x, y).
- Verify: We plugged the solution back into the original equations to ensure it was correct.
Practice Makes Perfect
Like any skill, mastering the combined method takes practice. The more problems you solve, the more comfortable you'll become with recognizing the best approach and executing the steps accurately. Don't be afraid to try different methods and see what works best for you.
Where can you find more practice problems? Textbooks, online resources, and worksheets are all great sources of practice problems. You can also create your own problems by changing the coefficients and constants in the equations.
Beyond the Basics
The combined method is a fundamental tool for solving systems of linear equations, but it's also a stepping stone to more advanced mathematical concepts. Understanding how to solve these equations is essential for topics like linear algebra, calculus, and even real-world applications in fields like engineering, economics, and computer science.
So, keep practicing, keep exploring, and keep building your math skills! You've got this!
Final Thoughts
Solving systems of equations might seem daunting at first, but with the right tools and a step-by-step approach, it becomes much more manageable. The combined method is a powerful technique that gives you the flexibility to tackle a variety of problems. Remember to understand the underlying concepts, practice regularly, and always verify your solutions. With dedication and perseverance, you'll become a master of solving equations!
