Solving Systems Of Linear Equations A Step-by-Step Guide
Hey guys! 👋 Feeling a bit tangled up with systems of linear equations? No worries, we've all been there! These sets of equations might seem intimidating at first, but with the right approach, they become quite manageable. This article will walk you through solving three different systems of linear equations, each with its unique quirks and solutions. We'll be tackling the problems step-by-step, so you can follow along and master these skills. Let's dive in!
System 1: 2x + y = 8 and x - y = 1
Linear equations are the backbone of many mathematical models, and understanding how to solve them is crucial. Our first system presents us with two equations:
- 2x + y = 8
- x - y = 1
There are a couple of ways we can approach this, but the elimination method often shines in scenarios like this one. Notice how the 'y' terms have opposite signs? That's our cue! If we add these two equations together, the 'y' terms will neatly cancel each other out, leaving us with an equation in just 'x'.
Let's do it! Adding equation (1) and equation (2), we get:
(2x + y) + (x - y) = 8 + 1
Simplifying this, we have:
3x = 9
Now, solving for 'x' is a breeze. Divide both sides by 3:
x = 3
Awesome! We've found the value of 'x'. But we're not done yet; we need to find 'y' as well. To do this, we can substitute the value of 'x' (which is 3) into either of the original equations. Let's use equation (2) because it looks simpler:
x - y = 1
Substitute x = 3:
3 - y = 1
To isolate 'y', subtract 3 from both sides:
-y = -2
Multiply both sides by -1 to get 'y' by itself:
y = 2
So, there you have it! The solution to this system of equations is x = 3 and y = 2. This means that the point (3, 2) is where these two lines intersect on a graph. We can always double-check our answer by plugging these values back into both original equations to make sure they hold true. Let's verify:
For equation (1): 2(3) + 2 = 6 + 2 = 8 (Correct!)
For equation (2): 3 - 2 = 1 (Correct!)
See? Everything checks out. Solving systems of linear equations is like detective work – you gather clues, follow the trail, and eventually uncover the solution. With practice, you'll become a pro at it!
System 2: x + 2y = 6 and x - y = 0
Moving on to our next challenge, we have another system of linear equations:
- x + 2y = 6
- x - y = 0
This time, let’s tackle this system using the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation. Looking at equation (2), we can easily solve for 'x':
x - y = 0
Add 'y' to both sides:
x = y
Perfect! We now know that 'x' is equal to 'y'. This makes the substitution step super simple. We'll substitute 'y' for 'x' in equation (1):
x + 2y = 6
Substitute x = y:
y + 2y = 6
Combine like terms:
3y = 6
Now, divide both sides by 3:
y = 2
Fantastic! We've found the value of 'y'. Since we know that x = y, finding 'x' is a piece of cake:
x = y = 2
So, the solution to this system is x = 2 and y = 2. This means the point of intersection for these two lines is (2, 2). Let's verify our solution by plugging these values back into the original equations:
For equation (1): 2 + 2(2) = 2 + 4 = 6 (Correct!)
For equation (2): 2 - 2 = 0 (Correct!)
Great! Our solution holds up. Substitution is a powerful technique, especially when one equation can be easily solved for one variable. It's all about finding the right tool for the job, and in this case, substitution fit the bill perfectly. Remember, practice makes perfect, so keep at it!
System 3: 3x + y = 9 and x - y = 3
Alright, let's jump into our third system of linear equations:
- 3x + y = 9
- x - y = 3
This system looks quite similar to our first one, doesn't it? We have 'y' terms with opposite signs, which makes the elimination method another excellent choice. Adding these two equations together should eliminate 'y' and leave us with an equation in 'x'.
Let's add equation (1) and equation (2):
(3x + y) + (x - y) = 9 + 3
Simplifying, we get:
4x = 12
Now, divide both sides by 4 to solve for 'x':
x = 3
Excellent! We've found 'x'. Now, let's substitute this value back into one of the original equations to find 'y'. We'll use equation (2) again because it seems simpler:
x - y = 3
Substitute x = 3:
3 - y = 3
Subtract 3 from both sides:
-y = 0
Multiply both sides by -1:
y = 0
There we have it! The solution to this system is x = 3 and y = 0. The point of intersection for these lines is (3, 0). As always, let's verify our solution by plugging these values back into the original equations:
For equation (1): 3(3) + 0 = 9 + 0 = 9 (Correct!)
For equation (2): 3 - 0 = 3 (Correct!)
Fantastic! Everything checks out perfectly. Mastering the elimination method can significantly simplify solving systems of equations, especially when variables have opposite signs. Each time you solve a system, you're honing your skills and building confidence. Keep practicing, and you'll become a true equations-solving whiz!
Conclusion
So, guys, we've journeyed through solving three different systems of linear equations using both the elimination and substitution methods. Remember, the key is to choose the method that best suits the problem and to always verify your solutions. Linear equations pop up everywhere in math and science, so mastering them is a fantastic investment in your skills. Keep practicing, stay curious, and you'll be solving complex problems in no time! You've got this! 💪
