Find The Equation Of A Line: Slope -2, Point (6,8)

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the fascinating world of mathematics, specifically tackling a common challenge: finding the equation of a line when you're given its slope and a point it passes through. This is a fundamental skill, and mastering it will unlock so many other mathematical concepts for you. So, let's get our hands dirty with a specific problem: What is the equation of a line with a slope of -2 that passes through the point (6,8)? We'll break down the steps, explain the reasoning, and make sure you're totally comfortable with this by the end. We'll explore the different forms of linear equations and how to switch between them, ensuring you're equipped for any scenario.

Understanding the Basics: Slope and Point-Slope Form

Before we jump into solving our specific problem, let's recap some core concepts. The slope of a line, often denoted by 'm', tells us how steep the line is and in which direction it's trending. A positive slope means the line goes upwards from left to right, while a negative slope, like our -2, means it goes downwards. The point (6,8) is just one of the infinite points that lie on this specific line. The key to solving this lies in the point-slope form of a linear equation, which is a lifesaver when you have a point and the slope. The general form of the point-slope equation is $y - y_1 = m(x - x_1)$, where 'm' is the slope and $(x_1, y_1)$ is the given point. This formula is derived from the definition of slope: m = rac{y_2 - y_1}{x_2 - x_1}. If we let $(x, y)$ be any point on the line and $(x_1, y_1)$ be our specific known point, then the slope between them must be 'm'. Rearranging this gives us the point-slope form. It's super handy because it directly incorporates the information we're given. So, for our problem, we have $m = -2$ and our point $(x_1, y_1) = (6, 8)$. Plugging these values directly into the point-slope formula, we get: $y - 8 = -2(x - 6)$. This is technically a valid equation of the line, but often, we're asked to express it in other forms, like the slope-intercept form ($y = mx + b$) or the standard form ($Ax + By = C$). Let's explore how to convert our point-slope equation into these more common formats.

Converting to Slope-Intercept Form ($y = mx + b$)

The slope-intercept form is probably the most frequently used way to represent a linear equation. It's incredibly useful because it clearly shows the slope ('m') and the y-intercept ('b') – the point where the line crosses the y-axis. Our goal here is to isolate 'y' on one side of the equation. Starting with our point-slope form: $y - 8 = -2(x - 6)$. The first step is to distribute the -2 on the right side of the equation: $y - 8 = -2x + (-2 imes -6)$. This simplifies to $y - 8 = -2x + 12$. Now, to get 'y' by itself, we need to add 8 to both sides of the equation: $y - 8 + 8 = -2x + 12 + 8$. This gives us our final equation in slope-intercept form: $y = -2x + 20$. See? It's pretty straightforward once you get the hang of distributing and isolating the variable. You can immediately see that the slope is indeed -2, and the y-intercept is 20. This means our line crosses the y-axis at the point (0, 20). It's awesome how much information you can glean from just one equation! Remember, the 'b' in $y = mx + b$ is specifically the y-coordinate of the y-intercept. Understanding these forms helps you visualize the line better and makes it easier to compare different lines.

Converting to Standard Form ($Ax + By = C$)

Another common way to express a linear equation is in standard form, which is typically written as $Ax + By = C$. In this form, A, B, and C are usually integers, and A is generally a positive integer. This form is particularly useful when dealing with systems of linear equations or graphing lines using intercepts. Let's take our slope-intercept form, $y = -2x + 20$, and convert it into standard form. The goal here is to get the x and y terms on the same side of the equation, with the constant term on the other. First, we want to move the $-2x$ term to the left side. To do this, we add $2x$ to both sides of the equation: $y + 2x = -2x + 20 + 2x$. This simplifies to $2x + y = 20$. Now, we check if this fits the criteria for standard form. A is 2 (positive integer), B is 1 (integer), and C is 20 (integer). So, yes, $2x + y = 20$ is the equation of our line in standard form. It's pretty neat how different forms can represent the exact same line, right? Each form has its own advantages depending on what you need to do with the equation. For instance, if you were asked to find the x-intercept, rearranging into standard form first often makes that calculation simpler: set y=0, so $2x = 20$, which means $x=10$. The x-intercept is (10,0).

Checking Our Work and The Options

Now, it's always a good idea to double-check our work, especially when presented with multiple-choice options. We found the equation of the line in slope-intercept form to be $y = -2x + 20$, and in standard form as $2x + y = 20$. Let's see how these match up with the options provided:

• $2x + y = 4$
• $y - 2x = 20$
• $y - 2x = 4$
• $2x + y = 20$

Our derived standard form equation, $2x + y = 20$, is directly listed as one of the options! This gives us high confidence in our solution. We can also quickly verify if the point $(6, 8)$ satisfies this equation: $2(6) + 8 = 12 + 8 = 20$. It does! This confirms our answer is correct. Let's quickly look at the other options. The equation $y - 2x = 20$ is equivalent to $y = 2x + 20$, which has a positive slope, not -2. The other options, $2x + y = 4$ and $y - 2x = 4$, don't satisfy the point (6,8) nor the slope requirement when rearranged. For example, $2x + y = 4$ would mean $y = -2x + 4$, which has the correct slope but a y-intercept of 4, not 20. So, by using the point-slope form, converting to slope-intercept and standard forms, and then verifying against the given point and options, we've confidently arrived at the correct answer.

Why This Matters: Real-World Applications

Guys, understanding how to find the equation of a line isn't just about acing math tests; it's a foundational skill with tons of real-world applications. Think about tracking the progress of something that changes at a constant rate – that's a linear relationship! For instance, if you're saving money, and you add $50 every week, the amount of money you have over time can be represented by a linear equation. The slope would be 50 (dollars per week), and the y-intercept would be the initial amount of money you started with. Similarly, in physics, concepts like velocity and displacement often involve linear relationships. If an object is moving at a constant speed, its distance traveled over time can be plotted as a straight line. The slope represents the speed, and the y-intercept represents the initial position. In economics, you might see linear models used to represent costs, revenues, or supply and demand curves. Even in computer graphics, lines and planes are fundamental building blocks. So, the next time you encounter a problem involving rates of change or linear relationships, remember these steps. You’re not just doing math; you’re learning a powerful tool for understanding and modeling the world around you. Keep practicing, and don't be afraid to experiment with different types of problems. The more you practice, the more intuitive it becomes, and soon you'll be spotting linear relationships everywhere!