Graph Of System Of Equations 2x+y=6 And 6x+3y=12
Hey guys! Today, we're diving deep into the fascinating world of systems of equations and their graphical representations. We'll be tackling a specific problem that asks us to identify the graph of the following system of equations:
The question presents us with four options: overlapping lines, parallel lines, intersecting lines, and a curve intersecting with a line. To crack this, we'll need to understand what each option represents and how to analyze the given equations. So, grab your thinking caps, and let's get started!
Understanding the Options
Before we jump into the solution, let's quickly review the different types of lines we might encounter in a system of equations:
- Overlapping Lines: Imagine two lines drawn perfectly on top of each other. They share every single point in common. In terms of equations, this means the two equations are essentially the same, just possibly multiplied by a constant. Think of it like this: one equation is just a scaled-up version of the other.
- Parallel Lines: These lines run side by side, never touching. They have the same slope but different y-intercepts. Picture railroad tracks stretching into the distance – that's the visual we're going for here. Equation-wise, the coefficients of x and y will be proportional, but the constant terms will not be.
- Intersecting Lines: These lines cross each other at a single point. This point represents the unique solution to the system of equations. The slopes of these lines are different. If you graph them, they'll meet at one specific spot, giving you the x and y values that satisfy both equations.
- A Curve Intersecting with a Line: This option involves a non-linear equation (the curve) and a linear equation (the line). Their intersection points represent the solutions to the system. We won't see this in our particular problem, but it's good to know it exists!
Analyzing the Equations
Now that we know our options, let's roll up our sleeves and examine the equations:
The key to figuring out the relationship between these lines lies in comparing their slopes and y-intercepts. There are a couple of ways we can do this. One popular method is to convert both equations into slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. Let's do that!
Converting to Slope-Intercept Form
Equation 1: 2x + y = 6
To isolate y, we subtract 2x from both sides:
y = -2x + 6
So, the slope of the first line (m₁) is -2, and the y-intercept (b₁) is 6.
Equation 2: 6x + 3y = 12
First, subtract 6x from both sides: 3y = -6x + 12
Now, divide both sides by 3 to get y by itself:
y = -2x + 4
Here, the slope of the second line (m₂) is -2, and the y-intercept (b₂) is 4.
Comparing Slopes and Y-Intercepts
Alright, we've got the slopes and y-intercepts for both lines. Let's compare them:
- Slope of line 1 (m₁) = -2
- Slope of line 2 (m₂) = -2
Notice anything? The slopes are the same! This tells us that the lines are either parallel or overlapping. Parallel lines have the same slope, and overlapping lines are essentially the same line, so they also have the same slope. So far so good. Next, let's compare Y intercepts:
- Y-intercept of line 1 (b₁) = 6
- Y-intercept of line 2 (b₂) = 4
Here is something of great importance, the y-intercepts are different! This is a game-changer. If the slopes were the same and the y-intercepts were also the same, the lines would be overlapping. But because the y-intercepts are different, the lines are parallel.
Alternative Method: Checking for Proportionality
Another way to approach this problem is to check if the equations are proportional. If one equation is a multiple of the other, the lines are either overlapping or parallel. Let's see if this holds true for our equations.
If we multiply the first equation (2x + y = 6) by 3, we get 6x + 3y = 18. Now, compare this to the second equation (6x + 3y = 12).
The left-hand sides (6x + 3y) are the same in both equations but the right-hand sides are not (18 vs. 12). This means the equations are not proportional, indicating that the lines are parallel.
Reaching the Conclusion
We've analyzed the equations using two different methods, and both point to the same conclusion: the lines are parallel. They have the same slope but different y-intercepts. Think of those railroad tracks again – they run alongside each other, never meeting.
Final Answer: The Graph of the System of Equations
Therefore, the graph of the system of equations is B. Parallel lines. We've cracked the code! We analyzed the slopes and y-intercepts, and we even checked for proportionality. By understanding the relationship between equations and their graphical representations, we were able to confidently arrive at the correct answer.
Key Takeaways
Before we wrap up, let's highlight some key takeaways from this problem:
- Slope-intercept form (y = mx + b) is your friend: It makes it super easy to identify the slope and y-intercept, which are crucial for determining the relationship between lines.
- Same slope, different y-intercepts = Parallel lines: This is a golden rule to remember when dealing with systems of equations.
- Same slope, same y-intercept = Overlapping lines: These lines are essentially the same.
- Different slopes = Intersecting lines: These lines will cross at a single point.
- Checking for proportionality can be a quick shortcut: If one equation is a multiple of the other, you're dealing with either parallel or overlapping lines.
Level Up Your Skills
Systems of equations might seem daunting at first, but with practice, they become much easier to handle. The more you work with them, the better you'll become at recognizing the patterns and applying the right techniques. So, keep practicing, keep exploring, and keep expanding your mathematical horizons! You guys got this!
Now, go forth and conquer more mathematical challenges! And remember, understanding the fundamentals is key to unlocking the complexities of math. Until next time, happy problem-solving!
