Line Equation: Point-Slope Form Explained
Finding the equation of a line that passes through two given points is a fundamental concept in algebra. In this comprehensive guide, we'll walk through the process step-by-step, focusing on expressing the answer in point-slope form. Specifically, we'll tackle the problem of finding the equation of the line that passes through the points (-9, -4) and (-5, -1). Let's dive in!
Understanding Point-Slope Form
Before we jump into the calculations, let's quickly recap what point-slope form is all about. The point-slope form of a linear equation is expressed as:
y - y1 = m(x - x1)
Where:
(x1, y1)is a known point on the line.mis the slope of the line.
The beauty of point-slope form is that it allows us to write the equation of a line using just a single point and the slope, making it incredibly useful when we have two points and need to find the equation.
Step 1: Calculate the Slope
The first thing we need to do is calculate the slope (m) of the line that passes through the points (-9, -4) and (-5, -1). The slope is a measure of the steepness of the line and is defined as the change in y divided by the change in x. The formula for calculating the slope (m) given two points (x1, y1) and (x2, y2) is:
m = (y2 - y1) / (x2 - x1)
In our case, we have:
(x1, y1) = (-9, -4)(x2, y2) = (-5, -1)
Plugging these values into the formula, we get:
m = (-1 - (-4)) / (-5 - (-9))
m = (-1 + 4) / (-5 + 9)
m = 3 / 4
So, the slope of the line that passes through the points (-9, -4) and (-5, -1) is 3/4.
Main keyword usage: We start by calculating the slope, a crucial step in finding the line equation.
Step 2: Use Point-Slope Form
Now that we have the slope (m = 3/4), we can use the point-slope form of the equation of a line. We have two points to choose from: (-9, -4) and (-5, -1). Let's use the point (-9, -4) as (x1, y1). Plugging the values into the point-slope form, we get:
y - (-4) = (3/4)(x - (-9))
y + 4 = (3/4)(x + 9)
This is the equation of the line in point-slope form using the point (-9, -4). We could also use the point (-5, -1) and arrive at an equivalent equation:
y - (-1) = (3/4)(x - (-5))
y + 1 = (3/4)(x + 5)
Both of these equations are correct and represent the same line. For simplicity, let's stick with the first one: y + 4 = (3/4)(x + 9).
Main keyword usage: The point-slope form is essential for defining the line equation. Using one of the given points and the slope we calculated, we can directly plug these values into the point-slope form formula.
Step 3: Simplify (Optional, but Recommended)
While the equation y + 4 = (3/4)(x + 9) is in point-slope form, it's often helpful to simplify it further, especially if you need to convert it to slope-intercept form (y = mx + b) or standard form (Ax + By = C). However, the question specifically asks for the answer in point-slope form, so simplification is not strictly necessary. But let's go ahead and do it for completeness. We can simplify it to slope-intercept form:
y + 4 = (3/4)x + (3/4)(9)
y + 4 = (3/4)x + 27/4
y = (3/4)x + 27/4 - 4
y = (3/4)x + 27/4 - 16/4
y = (3/4)x + 11/4
So, the slope-intercept form of the equation is y = (3/4)x + 11/4. However, remember that we were asked for the point-slope form. Therefore, we stick with the equation we found in Step 2: y + 4 = (3/4)(x + 9). Remember, point-slope form highlights a specific point on the line and its slope.
Main keyword usage: Although not explicitly required, simplification of the line equation can be beneficial. We can convert from point-slope form to slope-intercept form.
Alternative Approach: Using the Other Point
As mentioned earlier, we could have used the other point, (-5, -1), in the point-slope form. Let's see how that works out:
y - (-1) = (3/4)(x - (-5))
y + 1 = (3/4)(x + 5)
This is also a valid point-slope form of the equation. If we were to simplify this to slope-intercept form, we would get the same result as before (y = (3/4)x + 11/4). Let's quickly verify:
y + 1 = (3/4)x + (3/4)(5)
y + 1 = (3/4)x + 15/4
y = (3/4)x + 15/4 - 1
y = (3/4)x + 15/4 - 4/4
y = (3/4)x + 11/4
As you can see, regardless of which point we use, the slope-intercept form remains the same. The point-slope form, however, will look different depending on the point used, but they all represent the same line.
Main keyword usage: The line equation remains consistent, regardless of the point chosen for the point-slope form. Choosing either point results in the same line equation when converted to slope-intercept form.
Special Cases: Horizontal and Vertical Lines
It's worth mentioning the special cases of horizontal and vertical lines. If the slope is zero (m = 0), the line is horizontal. The equation of a horizontal line is of the form y = c, where c is a constant. This is because the y-value is the same for all points on the line.
If the slope is undefined (division by zero), the line is vertical. The equation of a vertical line is of the form x = c, where c is a constant. This is because the x-value is the same for all points on the line.
In our case, the slope is 3/4, which is neither zero nor undefined. Therefore, our line is neither horizontal nor vertical. If we had encountered a horizontal or vertical line, we would express the equation accordingly.
Main keyword usage: Understanding when a line equation results in horizontal or vertical lines is crucial. When the slope is zero, it's a horizontal line. When the slope is undefined, it's a vertical line.
Final Answer
The equation of the line that passes through the points (-9, -4) and (-5, -1) in point-slope form is:
y + 4 = (3/4)(x + 9)
Alternatively, you could also express it as:
y + 1 = (3/4)(x + 5)
Both of these equations are correct and represent the same line. Choose whichever form you prefer, or the one that best suits the context of the problem.
Key Takeaways
- To find the equation of a line in point-slope form, you need a point on the line and the slope of the line.
- The slope can be calculated using the formula
m = (y2 - y1) / (x2 - x1)given two points on the line. - The point-slope form of the equation is
y - y1 = m(x - x1). - You can use either of the given points in the point-slope form.
- Simplifying the equation to slope-intercept form (y = mx + b) can be helpful, but is not required if the question specifically asks for point-slope form.
- Be aware of the special cases of horizontal and vertical lines.
Guys, understanding these steps will help you confidently tackle any problem involving finding the equation of a line in point-slope form. Keep practicing, and you'll master this concept in no time! Remember, the line equation is a fundamental tool in algebra, and mastering it will open doors to more advanced topics.
Main keyword usage: Mastering the line equation in point-slope form is a fundamental concept in algebra. It is essential to understand how to calculate the slope and use the point-slope form formula to find the line equation.
Practice Problems
To solidify your understanding, try these practice problems:
- Find the equation of the line that passes through the points (2, 3) and (4, 7) in point-slope form.
- Find the equation of the line that passes through the points (-1, 5) and (2, -1) in point-slope form.
- Find the equation of the line that passes through the points (0, -2) and (3, 4) in point-slope form.
Good luck, and happy problem-solving!
Main keyword usage: Practice is key to mastering the line equation in point-slope form. Make sure to work through practice problems to reinforce your understanding of the slope and point-slope form formula.