Point-Slope Form Equation Passing Through (3, 2) With Slope 1/3

October 14, 2025 by Scholario Team 64 views

Let's dive into the point-slope form of a linear equation! If you've ever wondered how to express the equation of a line when you know a point it passes through and its slope, you're in the right place. This article will break down the concept, walk through an example, and help you understand how to use the point-slope form effectively. So, let's get started and make math a little less mysterious, shall we?

Understanding the Point-Slope Form

The point-slope form is a powerful tool in algebra for representing linear equations. It's particularly handy when you have a specific point on the line and the slope of the line. The point-slope form is given by:

*y - y₁ = m(x - x₁) *

Where:

(x₁, y₁) is a known point on the line.
m is the slope of the line.
x and y are the variables representing any other point on the line.

This form is derived directly from the definition of slope, which is the change in y divided by the change in x. When you rearrange the slope formula, you naturally arrive at the point-slope form. Understanding this connection can make it easier to remember and apply the formula. It's all about the relationship between the rise (change in y) and the run (change in x)! Using this point-slope form not only simplifies expressing linear equations but also provides a direct and intuitive way to visualize and interpret the line's characteristics on a coordinate plane. For example, you can quickly identify a specific point the line passes through and the steepness and direction of the line just by looking at the equation. This makes the point-slope form invaluable for various applications in mathematics and real-world problem-solving scenarios. So, keep this formula handy – you'll find it incredibly useful as you continue your journey in algebra and beyond!

Breaking Down the Components

To really get comfortable with the point-slope form, let's break down each component:

(x₁, y₁) - The Known Point: This is a specific point with known coordinates that the line passes through. It's your anchor point, the fixed spot you use to define the line's position in the coordinate plane. When you're given a problem, this point will typically be provided as an ordered pair, like (3, 2) or (-1, 4). This point is crucial because it grounds the line in a specific location, ensuring it passes through this exact coordinate. Recognizing and correctly substituting the x and y values of this point into the point-slope form is a fundamental step in constructing the equation of the line. Make sure you pay close attention to the signs of the coordinates, as they will directly affect the equation you derive. Think of this point as the starting place for drawing your line – it's where everything begins.
m - The Slope: The slope, often denoted as m, represents the steepness and direction of the line. It tells you how much the line rises (or falls) for every unit increase in the horizontal direction. A positive slope indicates that the line rises as you move from left to right, while a negative slope means the line falls. The numerical value of the slope gives you the rate of this rise or fall; a larger magnitude signifies a steeper line, and a smaller magnitude indicates a flatter line. For instance, a slope of 2 means that for every 1 unit you move to the right, the line goes up 2 units. Understanding the slope is vital because it determines the line's inclination and overall behavior. In the point-slope form, the slope m acts as a multiplier, affecting how the change in x relates to the change in y. Accurately identifying and substituting the slope value is essential for correctly expressing the line's orientation in the coordinate plane. So, keep a close eye on the slope – it’s a key indicator of the line's character!
(x, y) - The Variables: These are the variables that represent any other point on the line. They are essential for creating a general equation that describes all the points on the line, not just the specific point you started with. The variables x and y are placeholders that can take on any values that satisfy the equation, thereby tracing out the infinite number of points that lie on the line. Unlike x₁ and y₁, which are fixed coordinates of a known point, x and y are dynamic and can change. They maintain the relationship defined by the slope m and the known point (x₁, y₁). In the point-slope form equation, the x and y terms help to express the proportional relationship between the horizontal and vertical changes along the line. They allow you to find any point on the line by plugging in a value for one variable and solving for the other. Think of x and y as the moving pieces in the equation that define the line's path across the coordinate plane. They are what make the equation a versatile tool for describing the entire line, not just a single point.

Applying the Point-Slope Form: A Step-by-Step Guide

Let's walk through a step-by-step process to apply the point-slope form, so you can see exactly how it works. This will help you tackle any similar problems with confidence.

Identify the Given Information: Start by pinpointing the known point (x₁, y₁) and the slope m. These are your essential ingredients for the point-slope form recipe. Make sure you carefully extract these values from the problem statement. Sometimes, the information might be presented in a slightly disguised way, such as describing the slope verbally or giving you two points from which you need to calculate the slope first. Being able to accurately identify these key pieces of information is the foundation for successfully using the point-slope form. So, always begin by clearly defining what (x₁, y₁) and m are in the given problem. This initial step will set you on the right path to solving the equation.
Plug the Values into the Formula: Once you've identified (x₁, y₁) and m, the next step is to substitute these values into the point-slope form equation: y - y₁ = m(x - x₁). This involves replacing x₁, y₁, and m in the equation with their corresponding numerical values. Be meticulous during this step, paying close attention to the signs and ensuring that each value is placed correctly. A small mistake in substitution can lead to a completely different equation, so precision is key. After substituting, you'll have an equation that represents the line passing through the given point and having the specified slope. This equation is the point-slope form representation of the line, and it's ready for further manipulation if needed. So, take your time, double-check your substitutions, and make sure everything lines up perfectly!
Simplify (if needed): After substituting the values into the point-slope form, you might need to simplify the equation. This often involves distributing the slope m across the terms inside the parentheses and rearranging the equation to a more standard form, such as slope-intercept form (y = mx + b) or standard form (Ax + By = C). Simplification makes the equation easier to interpret and use for various purposes, like graphing or comparing with other linear equations. To simplify, first distribute the m across (x - x₁), then collect like terms and rearrange the equation as needed. The specific steps for simplification will depend on what form you want to achieve. For example, to get to slope-intercept form, you would isolate y on one side of the equation. While the point-slope form is perfectly valid on its own, simplifying can often provide additional insights into the line's characteristics and make it more versatile for different applications. So, don’t skip this step if simplification is required or beneficial!

Example: Finding the Equation

Let's tackle the question: What is the equation in point-slope form of the line passing through the point (3, 2) with a slope of 1/3?

Identify the Given Information:
- The point (x₁, y₁) is (3, 2).
- The slope m is 1/3.
Plug the Values into the Formula:

Substitute these values into the point-slope form equation:

y - y₁ = m(x - x₁)

y - 2 = (1/3)(x - 3)
The Answer:

The equation in point-slope form is y - 2 = (1/3)(x - 3).

Why is Point-Slope Form Useful?

The point-slope form is incredibly useful for several reasons:

Directly Uses Given Information: It allows you to construct the equation of a line directly from a point and a slope, which are common pieces of information given in problems.
Easy to Convert: It can be easily converted into other forms, such as slope-intercept form or standard form, depending on your needs.
Conceptual Understanding: It reinforces the relationship between slope and a point on the line, providing a solid foundation for understanding linear equations.

Common Mistakes to Avoid

To ensure you're using the point-slope form correctly, here are some common mistakes to watch out for:

Incorrectly Substituting Values: Double-check that you're placing the values of x₁, y₁, and m in the correct spots in the formula.
Sign Errors: Pay close attention to the signs, especially when subtracting negative numbers.
Forgetting to Distribute: If you need to simplify, make sure you distribute the slope m to both terms inside the parentheses.

Practice Problems

To solidify your understanding, try these practice problems:

Find the equation of the line in point-slope form that passes through the point (-1, 4) with a slope of -2.
What is the equation of the line that passes through the point (5, -3) and has a slope of 1/2?

Work through these problems, and you'll become even more confident in using the point-slope form.

Conclusion

The point-slope form is a fantastic way to represent linear equations when you have a point and a slope. It’s a powerful tool that simplifies many algebra problems. By understanding the formula, breaking down its components, and practicing its application, you'll be well-equipped to tackle any line equation challenges that come your way. So, keep practicing, and remember, every problem is just another step toward mastering math! Guys, you've got this! Now go out there and conquer those lines!