Is (2, 2) A Solution For The System Of Equations?
Hey guys! Today, we're diving into a fun little math problem to see if a specific point is a solution to a system of equations. It might sound a bit intimidating, but trust me, it's super straightforward once you get the hang of it. We're going to break it down step by step, so you'll be solving these like a pro in no time! We'll tackle the question: Is (2, 2) a solution of the system of equations: y = -4x + 6 and 3x + 4y = -2? Let's jump right in and figure this out!
Understanding Systems of Equations and Solutions
Okay, before we jump into the specific problem, let's make sure we're all on the same page about what a system of equations actually is and what it means for a point to be a solution. Think of a system of equations as a set of two or more equations that we're looking at together. These equations usually involve the same variables, like x and y, and we're trying to find values for those variables that make all the equations true at the same time. That's where the idea of a 'solution' comes in.
What exactly is a solution, then? A solution to a system of equations is simply a set of values for the variables that, when plugged into each equation, make the equation a true statement. In the case of two variables (like x and y), we often represent the solution as an ordered pair (x, y). This ordered pair represents a point on a coordinate plane. If that point lies on the line represented by each equation in the system, then it's a solution to the system.
Why is this important? Well, systems of equations pop up all over the place in real-world applications. From figuring out the break-even point for a business to modeling the trajectory of a rocket, understanding how to solve these systems is a valuable skill. We use systems of equations to model scenarios where multiple conditions or constraints need to be satisfied simultaneously. Solving these systems helps us find the specific values that meet all the required conditions.
Now, let's connect this to our specific problem. We have two equations: y = -4x + 6 and 3x + 4y = -2. We're given the point (2, 2) and asked if it's a solution to this system. In other words, we need to check if plugging in x = 2 and y = 2 into both equations makes them true. If it does, then (2, 2) is indeed a solution. If even one of the equations becomes false, then (2, 2) is not a solution. So, let’s roll up our sleeves and see what happens when we plug those numbers in!
Step-by-Step Verification: Is (2, 2) the Solution?
Alright, let's get down to business and see if the point (2, 2) is the real deal when it comes to solving our system of equations. We’ve got two equations to work with, so we’ll take it one step at a time to make sure we don't miss anything.
Equation 1: y = -4x + 6
First up, we have the equation y = -4x + 6. This is a linear equation in slope-intercept form, which is pretty neat because we can easily visualize it as a straight line on a graph. But for now, we're not graphing; we're substituting! We're going to plug in x = 2 and y = 2 into this equation and see if it holds true. Replacing x and y with their values, we get:
2 = -4(2) + 6
Now, let’s simplify the right side of the equation. We start by multiplying -4 by 2, which gives us -8. So, the equation becomes:
2 = -8 + 6
Next, we add -8 and 6, which results in -2. Our equation now looks like this:
2 = -2
Hold the phone! Does 2 equal -2? Nope, it certainly does not. This is a false statement. So, right off the bat, we've hit a snag. Since the point (2, 2) doesn't satisfy the first equation, we already know it can't be a solution to the entire system. But just to be thorough, and for a little extra practice, let's go ahead and check the second equation anyway.
Equation 2: 3x + 4y = -2
Now, let’s take a look at our second equation: 3x + 4y = -2. This is another linear equation, but it’s in standard form. We're going to do the same thing here – substitute x = 2 and y = 2 into the equation and see if it's a true statement. Plugging in the values, we get:
3(2) + 4(2) = -2
First, let's do the multiplication. 3 times 2 is 6, and 4 times 2 is 8. So, the equation becomes:
6 + 8 = -2
Now, we add 6 and 8, which gives us 14. Our equation now looks like this:
14 = -2
Well, well, well… 14 does not equal -2. This is another false statement. As we suspected, the point (2, 2) doesn't satisfy this equation either.
The Verdict
So, after carefully substituting x = 2 and y = 2 into both equations, we found that neither equation holds true. This means that the point (2, 2) is definitely not a solution to the system of equations. It's like trying to fit a square peg in a round hole – it just doesn't work!
Why (2, 2) Isn't a Solution: A Graphical Perspective
Okay, we've done the math and confirmed that (2, 2) isn't a solution to our system of equations. But let's take a step back and think about why this is the case, especially from a graphical point of view. This can give us a deeper understanding of what solutions to systems of equations really mean.
Lines on a Graph
Remember, each equation in our system (y = -4x + 6 and 3x + 4y = -2) represents a straight line when graphed on a coordinate plane. Every point on a line satisfies the equation of that line. That means if you pick any point on the line and plug its x and y coordinates into the equation, the equation will be true.
The Solution as an Intersection
The solution to a system of two linear equations is the point where the two lines intersect. Why? Because the point of intersection is the only point that lies on both lines. That means its coordinates satisfy both equations simultaneously. It's the one and only place where both equations are true at the same time.
Visualizing (2, 2)
So, what does this mean for our point (2, 2)? We found that when we plugged x = 2 and y = 2 into our equations, neither equation was true. Graphically, this means that the point (2, 2) does not lie on either of the lines represented by our equations. It's somewhere else on the coordinate plane, not on either line. Therefore, it can't be the point where the lines intersect, and it can't be a solution to the system.
Thinking it Through
Imagine drawing the two lines on a graph. One line has the equation y = -4x + 6, which means it has a slope of -4 and a y-intercept of 6. The other line is 3x + 4y = -2. You could rewrite this in slope-intercept form if you wanted to visualize it more easily, but the important thing is that it's another straight line. If you were to plot the point (2, 2) on the same graph, you'd see that it's not on either of these lines. It's off to the side, not part of either one. This visual representation reinforces the idea that (2, 2) is not a solution to the system.
By understanding this graphical perspective, we can see that solving systems of equations isn't just about manipulating numbers; it's about finding the point where multiple conditions are met simultaneously. It's a powerful concept with applications in many different fields. So, next time you're solving a system of equations, remember the lines and their intersection – it can make the whole process much clearer!
Final Answer: Is (2, 2) a Solution?
We've gone through the process step by step, guys, and we've reached our conclusion! We started with the system of equations:
y = -4x + 6 3x + 4y = -2
And we wanted to know if the point (2, 2) was a solution to this system. We carefully substituted x = 2 and y = 2 into both equations to see if they held true. Here’s what we found:
- Equation 1: y = -4x + 6
- Substituting gave us 2 = -4(2) + 6, which simplified to 2 = -2. This is a false statement.
- Equation 2: 3x + 4y = -2
- Substituting gave us 3(2) + 4(2) = -2, which simplified to 14 = -2. This is also a false statement.
Since the point (2, 2) did not satisfy either equation, we can confidently say:
Final Answer: (B) No
(2, 2) is not a solution to the system of equations y = -4x + 6 and 3x + 4y = -2. We explored this not just mathematically, but also graphically, understanding that a solution to a system of equations represents the intersection point of the lines. This makes it clear why a point that doesn't lie on either line can't be a solution. Keep practicing, and you'll master these systems in no time!
