Point-Slope Form: Equation Of A Line Through (18, 20)

Hey guys! Let's dive into how to write the equation of a line using the point-slope form. This is super useful when you know a point on the line and the slope, and in this case, we're dealing with a line that passes through the point (18, 20) with a specific slope. So, grab your pencils, and let's get started!

Understanding Point-Slope Form

Okay, first things first, what exactly is the point-slope form? The point-slope form of a linear equation is a way to express the equation of a line using the slope of the line and a specific point that the line passes through. It's written as:

y - y₁ = m(x - x₁)

Where:

• y and x are the variables representing the coordinates of any point on the line.
• (x₁, y₁) is a known point on the line.
• m is the slope of the line. The slope, often denoted as m, represents the steepness and direction of a line. It tells you how much the y-value changes for every unit change in the x-value. A positive slope means the line goes uphill from left to right, a negative slope means it goes downhill, a slope of zero means a horizontal line, and an undefined slope means a vertical line.

This form is super handy because it directly incorporates the slope and a point, making it easy to write the equation if you have those two pieces of information. You can think of the point-slope form as a blueprint for building the equation of a line. It takes the essential elements – a point and a slope – and combines them in a way that defines the line's unique path on a graph.

Now, why is this form so useful? Well, imagine you're given a point and the slope of a line, and you need to find its equation. You could try to use the slope-intercept form (y = mx + b), but you'd first have to calculate the y-intercept (b). The point-slope form bypasses this extra step, allowing you to plug in the values directly and get the equation. It's like having a shortcut in your mathematical toolkit! Another reason to love the point-slope form is its flexibility. It's not just for writing equations; it's also excellent for understanding the relationship between different forms of linear equations. You can easily convert from point-slope form to slope-intercept form or standard form, making it a versatile tool for various situations. Whether you're solving problems in algebra class or tackling real-world scenarios involving linear relationships, the point-slope form is a valuable concept to have in your arsenal. It's a fundamental building block for understanding linear equations and their applications.

Applying Point-Slope Form to Our Problem

In our case, we have the point (18, 20), which means x₁ = 18 and y₁ = 20. We also have a slope, but let's call it m for now since the prompt didn't give the value. Our goal is to plug these values into the point-slope form and simplify.

Let’s say, for example, the slope m = -2. This means that for every 1 unit increase in x, the y-value decreases by 2 units. A negative slope indicates that the line is decreasing or going downwards from left to right. Now that we have a specific value for m, we can plug it into the point-slope form equation, along with the given point (18, 20), to create the equation of the line. This will give us a clear and actionable equation that we can use to analyze and graph the line.

So, let’s go ahead and substitute these values into the point-slope form, y - y₁ = m(x - x₁). We will replace y₁ with 20, m with -2, and x₁ with 18. This gives us the equation:

y - 20 = -2(x - 18)

This equation now represents the line that passes through the point (18, 20) with a slope of -2. It’s in point-slope form, which is useful for quickly seeing the slope and a point on the line. However, it's often helpful to simplify this equation into other forms, such as slope-intercept form (y = mx + b) or standard form (Ax + By = C), depending on what we need to do with the equation. The point-slope form provides a solid foundation for further analysis and manipulation of the equation, allowing us to easily extract key information and solve related problems. Whether we're graphing the line, finding other points on the line, or comparing it with other lines, having the equation in point-slope form is a powerful starting point.

Simplifying the Equation

We've got the equation in point-slope form, which is a great start. But sometimes, we need the equation in a different format, like slope-intercept form (y = mx + b) which is super useful for graphing. So, let's simplify our equation: y - 20 = -2(x - 18)

To get to slope-intercept form, we need to isolate y on one side of the equation. The first step is to distribute the -2 on the right side:

y - 20 = -2x + 36

Remember, distributing means multiplying the -2 by both terms inside the parentheses: -2 times x is -2x, and -2 times -18 is +36. Distributive property is a key concept in algebra, and it’s used to simplify expressions by multiplying a term by a group of terms inside parentheses. In this case, by distributing the -2, we've eliminated the parentheses and made the equation easier to work with. This step is crucial because it allows us to separate the variables and constants, moving us closer to isolating y. Understanding and applying the distributive property correctly ensures that our equation remains balanced and accurate throughout the simplification process.

Next, to isolate y, we need to get rid of the -20 on the left side. We do this by adding 20 to both sides of the equation:

y - 20 + 20 = -2x + 36 + 20

This simplifies to:

y = -2x + 56

And there you have it! We've successfully converted our equation from point-slope form to slope-intercept form. In the slope-intercept form, y = -2x + 56, we can easily see that the slope m is -2, which we already knew, and the y-intercept b is 56. The y-intercept is the point where the line crosses the y-axis, and in this case, it’s at (0, 56). This form of the equation is incredibly useful for graphing the line because we can quickly plot the y-intercept and then use the slope to find other points on the line. For example, since the slope is -2, we know that for every 1 unit we move to the right on the graph, we move 2 units down. This makes it easy to sketch the line accurately.

Alternative Slopes and Equations

Now, what if the slope was different? Let’s explore how the equation changes with another slope value. For example, suppose the slope m was 1/2, a positive fraction. How would this alter our equation? With a positive slope, the line will rise from left to right, instead of falling as it did with the negative slope of -2.

Let's plug m = 1/2 into the point-slope form along with our point (18, 20):

y - 20 = (1/2)(x - 18)

Now, let’s simplify this to slope-intercept form. First, distribute the 1/2:

y - 20 = (1/2)x - 9

Next, add 20 to both sides:

y = (1/2)x + 11

So, with a slope of 1/2, the equation of the line is y = (1/2)x + 11. This line has a positive slope, meaning it goes upwards as you move from left to right. The y-intercept is 11, which means the line crosses the y-axis at the point (0, 11). Comparing this equation to the previous one with a slope of -2, we can see how the slope affects the direction and steepness of the line. A smaller slope, like 1/2, results in a less steep line compared to a larger slope like -2.

This exercise demonstrates how changing the slope alters the equation and the characteristics of the line. It highlights the flexibility and usefulness of the point-slope form in adapting to different scenarios. Understanding how to manipulate equations based on given points and slopes is a fundamental skill in algebra and geometry, and it opens the door to solving a wide range of problems involving linear relationships.

Key Takeaways

Alright guys, let's recap what we've learned:

1. Point-Slope Form: Remember the formula: y - y₁ = m(x - x₁). This is your go-to when you have a point and a slope.
2. Plugging in Values: Substitute the given point and slope into the formula. This is a straightforward process, but accuracy is key!
3. Simplifying: Don't be afraid to simplify the equation into slope-intercept form (y = mx + b) if needed. This makes graphing easier and gives you a clearer picture of the line.

Using the point-slope form to find the equation of a line is a valuable skill in algebra. It provides a direct method for writing linear equations when given a point and a slope, and it helps in understanding the relationship between different forms of linear equations. By mastering this concept, you'll be well-equipped to tackle more complex problems involving lines and their properties.

Keep practicing, and you'll become a pro at using point-slope form in no time! You got this! This skill not only helps in academic settings but also in practical situations where linear relationships are involved, such as in physics, engineering, and economics. Understanding how lines behave and how to represent them mathematically is a fundamental building block for more advanced concepts in these fields.