Equation Of A Line Passing Through (-1, 1) With Slope -9
Hey guys! Today, we're diving into the fascinating world of linear equations. We're going to take a look at a specific problem: finding the equation of a line that gracefully glides through the point (-1, 1) and possesses a slope of -9. But not just any equation, we're aiming for the slope-intercept form, which is a super useful way to represent linear equations. This form not only tells us where the line intersects the y-axis but also reveals its steepness (the slope). So, buckle up, and let's embark on this mathematical journey together!
Understanding Slope-Intercept Form
Before we jump into solving our specific problem, let's quickly brush up on the slope-intercept form. It's a fundamental concept, and having a solid grasp on it will make everything else much smoother. The slope-intercept form of a linear equation is expressed as:
y = mx + b
Where:
- 'y' represents the y-coordinate of any point on the line.
- 'x' represents the x-coordinate of any point on the line.
- 'm' is the slope of the line, indicating its steepness and direction.
- 'b' is the y-intercept, the point where the line crosses the y-axis.
The beauty of this form lies in its simplicity and the immediate information it provides. By simply looking at the equation, we can instantly identify the slope and the y-intercept, giving us a clear picture of the line's behavior. Now, let's see how we can leverage this knowledge to solve our problem.
Deciphering the Significance of Slope (m)
Let's delve a little deeper into what the slope, represented by 'm' in our equation, actually signifies. The slope is a measure of the line's inclination – how much it rises (or falls) for every unit it runs horizontally. Think of it as the steepness of a hill; a steeper hill has a larger slope. Mathematically, the slope is defined as the "rise over run," which is the change in the y-coordinate divided by the change in the x-coordinate between any two points on the line. A positive slope indicates that the line is ascending as you move from left to right, while a negative slope means the line is descending. A slope of zero represents a horizontal line, and an undefined slope indicates a vertical line.
Unveiling the Y-Intercept (b) and Its Role
Next up, let's talk about the y-intercept, denoted by 'b' in our slope-intercept form. The y-intercept is the point where the line intersects the y-axis. It's the value of 'y' when 'x' is equal to zero. Imagine the y-axis as a vertical number line; the y-intercept tells us exactly where our line crosses this number line. Visually, it's the point where the line starts its journey on the coordinate plane. Knowing the y-intercept is crucial because it anchors the line in a specific location, giving us a starting point to visualize and graph the entire line.
The Point-Slope Form: A Stepping Stone
While we ultimately want the slope-intercept form, sometimes it's easier to start with another form called the point-slope form. This form is particularly handy when we know a point on the line and the slope. The point-slope form is expressed as:
y - y1 = m(x - x1)
Where:
- '(x1, y1)' is a known point on the line.
- 'm' is the slope of the line.
This form directly incorporates the given information – the point and the slope – making it a convenient starting point for many linear equation problems. Once we have the equation in point-slope form, we can easily manipulate it to get it into the slope-intercept form we desire. Now, let's apply this to our specific problem!
Demystifying the Point-Slope Form
The point-slope form might seem a bit daunting at first, but it's actually quite intuitive once you understand its underlying logic. It's essentially a rearrangement of the slope formula, connecting the slope of a line to a specific point on that line. Think of it as a way to describe the line's direction and position simultaneously. The beauty of this form lies in its ability to handle situations where we don't immediately know the y-intercept but do have a point and the slope. It provides a direct pathway to constructing the equation of the line without having to solve for the y-intercept first. This makes it a valuable tool in our mathematical arsenal, especially when dealing with problems like the one we're tackling today.
The Interplay Between Point-Slope and Slope-Intercept Forms
Understanding the relationship between the point-slope and slope-intercept forms is key to mastering linear equations. The point-slope form acts as a bridge, allowing us to transition from knowing a point and the slope to the familiar slope-intercept form. It's like having a recipe that starts with raw ingredients (the point and slope) and guides us step-by-step to the final dish (the slope-intercept equation). The process involves simple algebraic manipulations – distributing the slope and isolating 'y' – to transform the equation from point-slope to slope-intercept form. This connection highlights the flexibility and interconnectedness of different equation forms in mathematics, allowing us to choose the most convenient starting point based on the information we have.
Applying the Point-Slope Form to Our Problem
Okay, let's get our hands dirty and apply the point-slope form to our problem. We know the line passes through the point (-1, 1) and has a slope of -9. So, we have:
- x1 = -1
- y1 = 1
- m = -9
Plugging these values into the point-slope form, we get:
y - 1 = -9(x - (-1))
Simplifying the equation, we have:
y - 1 = -9(x + 1)
Great! We've successfully applied the point-slope form. Now, the next step is to transform this equation into the slope-intercept form. Are you ready? Let's do it!
The Art of Substitution in Action
The step we just performed, substituting the known values into the point-slope form, is a powerful technique in mathematics. Substitution allows us to replace variables with their corresponding values, effectively translating abstract concepts into concrete equations. It's like filling in the blanks in a template, using the given information to create a specific representation of the problem. In this case, we took the general point-slope form and customized it with our specific point and slope, resulting in an equation that accurately describes our line. This process of substitution is a cornerstone of problem-solving in mathematics and is used extensively across various topics.
The Importance of Careful Simplification
After substituting the values, we took a moment to simplify the equation. This step is crucial in ensuring accuracy and clarity. Simplification involves performing basic algebraic operations, such as distributing and combining like terms, to make the equation more manageable and easier to understand. It's like decluttering a workspace, removing unnecessary complexity to reveal the underlying structure. In our case, simplifying the equation made it clearer and prepared it for the final transformation into slope-intercept form. Neglecting simplification can lead to errors and make subsequent steps more challenging, highlighting the importance of this often-overlooked step.
Transforming to Slope-Intercept Form
To transform our equation into the slope-intercept form (y = mx + b), we need to isolate 'y' on one side of the equation. Let's start by distributing the -9 on the right side:
y - 1 = -9x - 9
Now, add 1 to both sides to isolate 'y':
y = -9x - 9 + 1
Simplifying, we get:
y = -9x - 8
Voilà ! We have successfully transformed the equation into slope-intercept form. Our final equation is y = -9x - 8.
Mastering the Distributive Property
The act of distributing the -9 in our equation highlights the significance of the distributive property in algebra. This property allows us to multiply a single term by multiple terms within parentheses, effectively expanding the expression. It's like unwrapping a package, revealing the individual components inside. In our case, distributing the -9 allowed us to remove the parentheses and pave the way for isolating 'y'. The distributive property is a fundamental tool in algebraic manipulation and is essential for simplifying expressions and solving equations. Without it, many algebraic operations would be significantly more challenging.
The Power of Isolating Variables
The core of transforming the equation into slope-intercept form lies in the concept of isolating a variable. This technique involves manipulating the equation to get the desired variable (in this case, 'y') alone on one side. It's like solving a puzzle, strategically moving pieces around until you achieve the desired arrangement. To isolate 'y', we used inverse operations – adding 1 to both sides to undo the subtraction of 1. This process of isolating variables is a fundamental skill in algebra and is crucial for solving equations and understanding relationships between variables. It's the key to unlocking the solution and revealing the underlying truth of the equation.
The Final Equation and Its Interpretation
So, our final equation in slope-intercept form is y = -9x - 8. This equation tells us a lot about the line. The slope, 'm', is -9, which means the line is quite steep and slopes downwards from left to right. For every 1 unit we move to the right on the x-axis, the line drops 9 units on the y-axis. The y-intercept, 'b', is -8, which means the line crosses the y-axis at the point (0, -8).
The Visual Story of Slope and Y-Intercept
Our final equation, y = -9x - 8, is more than just a string of symbols; it tells a visual story of the line's behavior on the coordinate plane. The slope of -9 paints a picture of a steep descent, while the y-intercept of -8 anchors the line to a specific point on the y-axis. Imagine starting at the point (0, -8) and then moving one unit to the right and nine units down – you'll find yourself on the line again. This interplay between slope and y-intercept allows us to quickly visualize the line's direction, steepness, and position. It's a powerful example of how algebraic equations can translate into geometric representations, bridging the gap between abstract concepts and concrete visuals.
The Versatility of Slope-Intercept Form
The reason we strive to express linear equations in slope-intercept form is its versatility. It not only provides a clear snapshot of the line's slope and y-intercept but also makes it incredibly easy to graph the line. By simply plotting the y-intercept and then using the slope to find another point, we can draw the entire line with ease. Furthermore, the slope-intercept form is readily comparable; we can quickly compare the slopes and y-intercepts of different lines to understand their relative positions and orientations. This makes it a valuable tool for analyzing linear relationships and making predictions based on linear models.
Conclusion
And there you have it! We've successfully navigated the problem, found the equation of the line in slope-intercept form, and interpreted its meaning. We started with a point and a slope, used the point-slope form as a stepping stone, and transformed it into the elegant slope-intercept form. Remember, guys, the key to mastering these concepts is practice, practice, practice! So, keep exploring, keep solving, and keep having fun with math!
