Find Line Equation: Point (1,1), Slope -5

Hey guys! Today, we're diving into the super useful world of mathematics, specifically how to find the equation of a line when you've got a point it goes through and its slope. This is a fundamental concept, and once you get the hang of it, you'll see it everywhere. Let's tackle the specific problem: what is the equation of the line that passes through the point (1, 1) and has a slope of -5? This isn't just some abstract math problem; understanding this helps in graphing, understanding relationships between variables, and so much more. We'll break it down step-by-step, making sure everyone can follow along, whether you're a math whiz or just starting to explore.

Understanding the Building Blocks: Slope and Points

Before we jump into solving our specific problem, let's quickly recap what we're dealing with. In mathematics, a line is a fundamental geometric object. It's a straight, one-dimensional figure that has no thickness and extends infinitely in both directions. When we talk about the 'equation of a line,' we're essentially creating a mathematical statement that describes all the points that lie on that specific line. This equation acts like a rule or a blueprint for the line. Now, two key pieces of information usually define a unique line: its slope and a point it passes through. The slope (often denoted by the letter 'm') tells us how steep the line is and in which direction it's heading. A positive slope means the line goes upwards from left to right, while a negative slope means it goes downwards. A slope of zero indicates a horizontal line, and an undefined slope (which we usually represent with a vertical line) means the line goes straight up and down. The point is simply any coordinate (x, y) that is guaranteed to be on the line. Our specific problem gives us this crucial information: the line passes through the point (1, 1), and its slope is -5. This means for every one unit we move to the right along the x-axis, the line drops 5 units down on the y-axis. This relationship is key to understanding how the line behaves.

The Point-Slope Form: Your Best Friend

When you're asked to find the equation of a line given a point and its slope, the point-slope form of a linear equation is your absolute best friend. Seriously, guys, this form is designed precisely for this situation! It's incredibly straightforward. The general formula for the point-slope form is: y - y₁ = m(x - x₁). Here's what each part means: 'y' and 'x' are the general coordinates of any point on the line. 'm' is the slope of the line, which we already know is -5 in our case. 'x₁' and 'y₁' are the coordinates of the specific point that the line passes through. We are given this point as (1, 1), so x₁ = 1 and y₁ = 1. See? It all fits perfectly! This form directly incorporates the given information, making the calculation a breeze. It's called 'point-slope' because, well, it uses the point and the slope! It's a direct translation of the geometric definition of a line into an algebraic equation. Memorizing this form will save you a ton of time and confusion when tackling these types of problems. It's the most intuitive way to start when you have these two specific pieces of data.

Plugging in the Values: Let's Solve!

Alright, team, let's get down to business and solve our problem using the point-slope form. We have our formula: y - y₁ = m(x - x₁). We know our slope, m = -5, and our point is (x₁, y₁) = (1, 1). Now, we just need to substitute these values into the formula.

First, let's substitute the coordinates of the point (1, 1): y - 1 = m(x - 1)

Next, let's substitute the slope, m = -5: y - 1 = -5(x - 1)

And there you have it! This equation, y - 1 = -5(x - 1), is the equation of the line that passes through the point (1, 1) and has a slope of -5, expressed in point-slope form. It's that simple! This equation holds true for every single point that lies on this particular line. If you pick any point that's on this line, its x and y coordinates will satisfy this equation. This is the beauty of algebraic representation – it captures the essence of the geometric object. We've successfully used the given information to construct a precise mathematical description of our line. It’s a direct application of the formula, and by understanding the formula, the process becomes quite clear.

Transforming to Slope-Intercept Form (The Usual Suspect)

While the point-slope form is perfectly correct, mathematicians and educators often prefer to express linear equations in the slope-intercept form. This form is super recognizable and makes it easy to immediately see the slope and the y-intercept of the line. The slope-intercept form looks like this: y = mx + b. Here, 'm' is, as always, the slope, and 'b' is the y-intercept – the point where the line crosses the y-axis (where x = 0). So, let's take our point-slope equation and convert it.

We start with: y - 1 = -5(x - 1)

Our goal is to isolate 'y' on one side of the equation. First, we need to distribute the -5 on the right side: y - 1 = -5 * x + (-5) * (-1) y - 1 = -5x + 5

Now, to get 'y' by itself, we need to add 1 to both sides of the equation: y - 1 + 1 = -5x + 5 + 1 y = -5x + 6

Voila! The equation of the line in slope-intercept form is y = -5x + 6. From this form, we can instantly see that the slope 'm' is indeed -5, and the y-intercept 'b' is 6. This means the line crosses the y-axis at the point (0, 6). Converting between forms is a vital skill, as different contexts might require different representations. This transformation shows us how the initial point and slope dictate not only the line's steepness but also where it intersects the vertical axis. It's like uncovering more secrets about our line!

Verification: Does it all Add Up?

To be absolutely sure we've got the right equation, let's do a quick check. We need to verify two things:

1. Does the line y = -5x + 6 have a slope of -5?
2. Does the point (1, 1) lie on this line?

Let's start with the slope. In the slope-intercept form y = mx + b, the coefficient of 'x' is the slope 'm'. In our equation, y = -5x + 6, the coefficient of 'x' is -5. So, yes, the slope is -5. That checks out!

Now, let's check if the point (1, 1) satisfies the equation. We'll substitute x = 1 and y = 1 into our equation y = -5x + 6: 1 = -5(1) + 6 1 = -5 + 6 1 = 1

Since the equation holds true (1 equals 1), the point (1, 1) does indeed lie on the line represented by y = -5x + 6. This verification step is super important in math, guys. It builds confidence in your answer and helps catch any silly mistakes you might have made during the calculation. Seeing that both conditions are met confirms that our derived equation is correct. We've successfully found the unique line that fits the given criteria.

Real-World Applications: Why Does This Matter?

So, why do we even bother learning about equations of lines, slope, and points? It might seem like abstract stuff, but trust me, these concepts are the backbone of understanding many real-world phenomena. Think about physics, for example. If you're studying the motion of an object, its velocity is essentially the slope of its position-time graph. A constant velocity means a straight line with a constant slope! In economics, linear equations can model relationships between supply and demand, or cost and revenue. For instance, a business might use a linear equation to represent the cost of producing a certain number of items, where the slope indicates the cost per item (marginal cost). In engineering, you'll see linear relationships everywhere, from calculating electrical resistance to structural load calculations. Even in everyday life, if you're tracking how much money you spend over time, and you spend a consistent amount each day, that spending pattern can be represented by a line. The slope would be your daily spending rate. Understanding these mathematical relationships allows us to predict, analyze, and control various systems. So, the next time you're solving for the equation of a line, remember you're not just doing homework; you're gaining a powerful tool for understanding the world around you.

Conclusion: Mastering Linear Equations

In conclusion, finding the equation of a line given a point and a slope is a fundamental skill in mathematics. We started with the problem: what is the equation of the line that passes through the point (1, 1) and has a slope of -5? By utilizing the point-slope form (y - y₁ = m(x - x₁)), we directly substituted our given values to get y - 1 = -5(x - 1). We then transformed this into the more common slope-intercept form (y = mx + b) by performing algebraic manipulations, arriving at y = -5x + 6. We verified our answer by checking the slope and ensuring the given point satisfied the equation. This process not only solidifies your understanding of linear equations but also equips you with a valuable analytical tool. So, keep practicing, guys! The more you work with these concepts, the more intuitive they become, and the more you'll appreciate the elegance and power of mathematics in describing our world.