Solving Linear Equations And Completing Tables A Step-by-Step Guide

Hey guys! Today, we're diving into the world of linear functions and table completion. It might sound a bit daunting at first, but trust me, it's super manageable once we break it down. We've got a function, $-2y + x = 6$, and a domain (that's just a fancy word for the set of x-values we're using) of {4, 2, 0, -1}. Our mission? Complete a table by finding the corresponding y-values for these x-values. Oh, and the first step? We need to solve the equation for y. Let's get started!

Solving for y: The First Step to Table Completion

So, we're given the equation $-2y + x = 6$. To solve for y, we need to isolate it on one side of the equation. Think of it like a puzzle – we're moving pieces around until we get y all by itself.

First things first, let's get rid of that x term. We can do this by subtracting x from both sides of the equation. Remember, whatever we do to one side, we gotta do to the other to keep things balanced. This gives us:

$-2y + x - x = 6 - x$

Simplifying, we get:

$-2y = 6 - x$

Now, we're almost there! We have -2 multiplied by y, and we want just y. The opposite of multiplication is division, so let's divide both sides of the equation by -2:

$\frac{-2y}{-2} = \frac{6 - x}{-2}$

This simplifies to:

$y = \frac{6 - x}{-2}$

But wait, we can make this look a bit cleaner. Dividing by a negative is the same as multiplying by a negative, so let's distribute that negative sign in the denominator:

$y = \frac{6 - x}{-2} = \frac{-(6 - x)}{2} = \frac{x - 6}{2}$

Ta-da! We've solved for y! Our equation is now in the form $y = \frac{x - 6}{2}$. This means that for any x-value we plug in, we can easily find the corresponding y-value. This is a crucial step, guys. Solving for y makes the rest of the process smooth sailing.

Now that we've conquered this initial hurdle, we're ready to roll up our sleeves and actually plug in our x-values from the domain to complete the table. This is where the fun really begins, as we see how the equation translates into real numerical pairs. We're not just dealing with abstract symbols anymore; we're making concrete connections between x and y. It's like building the bridge that connects the input and output worlds of our function.

Understanding this process is fundamental to grasping the behavior of linear functions. Each x-value has a unique y-value partner, and the equation we solved for acts as the matchmaker, ensuring that every input finds its perfect output. This pairing principle is the bedrock upon which all linear functions are built. So, with our equation in hand, we're now equipped to populate our table and witness this pairing in action, turning our abstract formula into a tangible set of coordinates that we can even plot on a graph later on.

Completing the Table: Plugging in Our x-Values

Now for the fun part – plugging in our x-values and seeing what y-values pop out! Remember our domain is {4, 2, 0, -1}, and our equation is $y = \frac{x - 6}{2}$. Let's take each x-value one by one and substitute it into the equation.

1. When x = -1:

$y = \frac{-1 - 6}{2} = \frac{-7}{2} = -3.5$

So, when x is -1, y is -3.5. Cool!

2. When x = 4:

$y = \frac{4 - 6}{2} = \frac{-2}{2} = -1$

When x is 4, y is -1. We're on a roll!

3. When x = 2:

$y = \frac{2 - 6}{2} = \frac{-4}{2} = -2$

When x is 2, y is -2. Getting the hang of it?

4. When x = 0:

$y = \frac{0 - 6}{2} = \frac{-6}{2} = -3$

And finally, when x is 0, y is -3. Awesome!

We've now found the y-values that correspond to each x-value in our domain. This process of substitution is the heart and soul of evaluating functions. It's the engine that drives the relationship between input and output. By plugging in different x-values, we're essentially exploring the function's landscape, uncovering the y-values that define its shape and behavior.

Each pair of (x, y) values we've calculated represents a point on the line defined by our equation. These points are like the coordinates on a treasure map, guiding us along the function's path. The more points we find, the clearer the picture becomes. We're not just crunching numbers here; we're building a visual representation of the function, one point at a time. This understanding is crucial for grasping the concept of graphing linear equations, which is a fundamental skill in algebra and beyond.

But the beauty of this process extends beyond just finding points. It's about understanding the cause-and-effect relationship between x and y. Every change in x leads to a predictable change in y, dictated by the equation. This predictability is what makes linear functions so powerful and so widely used in various fields, from science to economics. So, as we fill in the table, we're not just ticking off boxes; we're gaining a deeper appreciation for the intricate dance between input and output that defines a function.

Filling in the Table and Discussing the Results

Okay, let's put all our findings into the table. Here's what we've got:

Now, let's fill in the table format provided:

From our calculations, when y = -3, x = 0. So we can fill that in too:

Let's take a moment to discuss what this table tells us. Each row represents a solution to the equation $-2y + x = 6$. In other words, if we plug in the x and y values from any row into the equation, it will hold true. This is the fundamental concept of a solution to an equation.

Furthermore, because this is a linear equation, these points would all fall on a straight line if we were to graph them. This is a key characteristic of linear functions – their solutions form a straight line. This visual representation makes understanding the relationship between x and y even more intuitive. We can see the constant rate of change, the slope, playing out visually as the line rises or falls.

The table also highlights the relationship between the domain and the range of the function. The domain, as we mentioned earlier, is the set of possible x-values, while the range is the set of corresponding y-values. Our table shows how each x-value in the domain maps to a specific y-value in the range. This mapping is the essence of a function – a rule that assigns a unique output to each input.

Moreover, the table allows us to make predictions about the function's behavior. We can see the trend – as x increases, y generally increases as well. This gives us a sense of the function's slope, whether it's increasing or decreasing. While the table only shows a few points, it gives us a valuable snapshot of the overall function and its characteristics. It's like a miniature model of the function's vast landscape, allowing us to explore its terrain and anticipate its twists and turns.

Solve for y in the equation -2y + x = 6, given the domain {4, 2, 0, -1}. Complete the table with the corresponding y values.

Solving Linear Equations and Completing Tables A Step-by-Step Guide