Solving 2 - 9x = X/2 - 1: Which System Of Equations?
Hey guys! Let's dive into solving the equation 2 - 9x = x/2 - 1 by figuring out which system of equations we can graph. It might sound a bit tricky at first, but trust me, we'll break it down step by step so it's super easy to understand. We will explore how to transform a single equation into a system of equations that can be graphed to find the solution. This method is not only useful for this specific problem but also for tackling various algebraic challenges. So, buckle up, and let's get started!
Understanding the Problem
Okay, so the main goal here is to figure out how to turn the given equation, 2 - 9x = x/2 - 1, into a set of equations that we can actually graph. When we talk about graphing equations, we're usually thinking about plotting them on a coordinate plane, which means we need equations in the form of y = something. Basically, we want to see where two lines intersect, and that intersection point will be the solution to our original equation.
To kick things off, remember that a system of equations is just a set of two or more equations that we solve together. Graphing is a visual way to solve these systems. Each equation represents a line, and the point where the lines cross (intersect) is the solution that satisfies both equations. This point gives us the x and y values that make both equations true.
Now, let's think about our single equation, 2 - 9x = x/2 - 1. It looks a bit different from the y = something form we need for graphing. But don't worry! Our mission is to rewrite this equation in a way that gives us two separate equations, each in the y = form. This is where the magic happens – we're essentially splitting one equation into two so we can graph them as lines and find their intersection point. This intersection point will visually represent the solution to the original equation.
Transforming the Equation
The heart of solving this problem lies in understanding how to transform the single given equation into a system of two equations. This involves a clever trick: we introduce a new variable, y, and create two separate equations, each expressing y in terms of x. This might sound a bit abstract, but it’s a powerful technique that allows us to use graphing as a problem-solving tool.
The original equation is 2 - 9x = x/2 - 1. To create a system of equations, we can set each side of the equation equal to y. This gives us two equations:
- y = 2 - 9x
- y = x/2 - 1
Why does this work? Well, at the solution point, both sides of the original equation are equal. By setting each side equal to y, we are essentially saying that at the solution, both expressions (2 - 9x and x/2 - 1) will have the same y-value. When we graph these two equations, the point where they intersect is where their y-values are equal, which corresponds to the solution of the original equation. Isn't that neat?
So, we've successfully transformed our single equation into a system of two equations. Each equation represents a straight line when graphed, and the solution to the original equation is represented by the point where these lines intersect. This is a fundamental concept in algebra and a visual way to understand solutions to equations.
Identifying the Correct System
Alright, now that we've massaged our original equation into a system of equations, the next step is to pinpoint the correct system from the options provided. Remember, the system we derived was:
- y = 2 - 9x
- y = x/2 - 1
To make sure we nail this, let's quickly recap what we did. We started with 2 - 9x = x/2 - 1 and split it into two separate equations by setting each side equal to y. This gave us two equations that we can graph as straight lines.
Now, let's imagine graphing these two lines. The first line, y = 2 - 9x, has a y-intercept of 2 (that's where the line crosses the y-axis) and a slope of -9 (it goes down steeply from left to right). The second line, y = x/2 - 1, has a y-intercept of -1 and a slope of 1/2 (it goes up gently from left to right).
The point where these lines intersect is the solution to the system, and the x-coordinate of this point is also the solution to our original equation, 2 - 9x = x/2 - 1. Cool, huh?
Now, all we need to do is compare the systems of equations given in the options to the one we derived. It should be a pretty straightforward match! This step is crucial to make sure we understand the connection between the algebraic manipulation and the visual representation of the solution.
Analyzing the Given Options
Time to put on our detective hats and carefully examine the options presented to us. Our goal is to match the system of equations we derived (y = 2 - 9x and y = x/2 - 1) with the correct choice. This is a critical step to ensure we understand how the algebraic manipulation translates to the correct graphical representation.
Let's take a look at each option and compare it to our system:
A. y = 2 - 9x y = x/2 - 1
B. y = -9x y = x/2
C. y = 9x y = x/2
Option A looks promising because it contains the equations y = 2 - 9x and y = x/2 - 1, which are exactly the equations we derived. This is a strong indicator that Option A is the correct answer. But, just to be thorough, let's quickly analyze the other options.
Option B has y = -9x and y = x/2. These equations are similar but not quite the same as ours. They're missing the constant terms (2 and -1), which are crucial for determining the correct y-intercepts of the lines. Without these constants, the lines would intersect at a different point, leading to a wrong solution.
Option C has y = 9x and y = x/2. This option changes the sign of the 9x term, which would significantly alter the slope of the line. Again, this would lead to a different intersection point and an incorrect solution. By carefully comparing each option, we can confidently identify the one that matches our derived system of equations.
Conclusion: Picking the Right Answer
Okay, guys, we've done the hard work of transforming the equation, understanding the system of equations, and analyzing the options. Now comes the satisfying part – choosing the correct answer! Remember, our original equation was 2 - 9x = x/2 - 1, and we transformed it into the following system of equations:
- y = 2 - 9x
- y = x/2 - 1
We meticulously compared this system to the options provided, and it became clear that:
Option A: y = 2 - 9x and y = x/2 - 1 is the correct answer.
Why? Because it perfectly matches the system we derived. The other options had slight variations that would change the slopes and intercepts of the lines, leading to an incorrect solution.
So, there you have it! We've successfully identified the system of equations that can be graphed to find the solution to 2 - 9x = x/2 - 1. This process highlights the power of transforming equations and using graphs to visualize solutions. Great job, everyone!
This problem demonstrates a powerful technique in algebra: converting a single equation into a system of equations to solve it graphically. By setting each side of the original equation equal to y, we create two linear equations. The intersection point of these lines on a graph represents the solution to the original equation. This method is particularly useful for visualizing solutions and understanding how different equations relate to each other. Understanding this concept opens doors to solving more complex problems and provides a solid foundation for further mathematical explorations. Keep practicing, and you'll become a pro at solving equations graphically!