Line Equation: Point (2, 8) And Slope 3/2

Have you ever wondered how to determine the equation of a line when you know a specific point it passes through and its slope? It's a fundamental concept in mathematics, especially in algebra and coordinate geometry. In this article, we'll explore a straightforward method to find the equation of a line given a point and a slope. We'll break down the process step by step, making it easy to understand and apply. Let's dive in and unlock the secrets of linear equations!

Understanding the Basics: Slope-Intercept Form

The slope-intercept form is a common way to represent a linear equation. It's written as:

y = mx + b

Where:

• y represents the y-coordinate of any point on the line.
• m represents the slope of the line, indicating its steepness and direction.
• x represents the x-coordinate of any point on the line.
• b represents the y-intercept, which is the point where the line crosses the y-axis.

The slope-intercept form provides a clear and concise way to understand the characteristics of a line. The slope (m) tells us how much the line rises or falls for every unit increase in x, while the y-intercept (b) gives us a fixed point on the line. Mastering this form is crucial for understanding and manipulating linear equations. We'll use this knowledge as a foundation for finding the equation of a line when given a point and a slope.

To truly grasp the concept, imagine a line on a graph. The slope is like the incline of a hill – a steeper slope means a faster climb or descent. The y-intercept is the starting point of your journey on that hill, the spot where you first step onto the line from the vertical axis. By understanding these components, you can visualize and manipulate lines with confidence. In the following sections, we'll apply this understanding to solve a specific problem: finding the equation of a line given a point and its slope.

The Point-Slope Form: A Powerful Tool

While the slope-intercept form is useful, another form, called the point-slope form, is particularly helpful when you know a point on the line and the slope. The point-slope form is written as:

y - y₁ = m(x - x₁)

Where:

• m is the slope of the line.
• (x₁, y₁) is a known point on the line.

The point-slope form is derived directly from the definition of slope. It essentially captures the relationship between any point (x, y) on the line and a specific point (x₁, y₁) that you already know. This form allows you to plug in the known values of the slope and the point's coordinates and then easily manipulate the equation to find the slope-intercept form if desired.

The beauty of the point-slope form lies in its direct applicability to situations where you have a point and a slope. Instead of having to solve for the y-intercept separately, you can directly substitute the given values into the formula and start simplifying. This makes it a powerful tool for tackling problems involving linear equations. In the next section, we'll demonstrate how to use the point-slope form to solve our specific problem: finding the equation of a line that passes through the point (2, 8) and has a slope of 3/2. By applying this form, you'll see how easily we can arrive at the desired equation.

Applying the Point-Slope Form: Our Example

Let's apply the point-slope form to find the equation of the line that passes through the point (2, 8) and has a slope of 3/2. Here's how we do it:

1. Identify the values:
   • m = 3/2 (the slope)
   • (x₁, y₁) = (2, 8) (the given point)
2. Substitute the values into the point-slope form:
   • y - 8 = (3/2)(x - 2)

Now we have the equation in point-slope form. To get it into the more familiar slope-intercept form (y = mx + b), we need to simplify and rearrange the equation. This involves distributing the slope and isolating y. The following steps will guide you through the process of converting from point-slope form to slope-intercept form, giving you a complete understanding of how to express the equation of a line in different ways. This skill is essential for various mathematical applications and will enhance your ability to work with linear equations.

Converting to Slope-Intercept Form: Simplifying the Equation

To convert the equation from point-slope form to slope-intercept form, we need to simplify and rearrange it. Let's continue from where we left off:

y - 8 = (3/2)(x - 2)

1. Distribute the slope (3/2) on the right side:
   • y - 8 = (3/2)x - (3/2)(2)
   • y - 8 = (3/2)x - 3
2. Isolate y by adding 8 to both sides:
   • y - 8 + 8 = (3/2)x - 3 + 8
   • y = (3/2)x + 5

Now we have the equation in slope-intercept form: y = (3/2)x + 5. This form clearly shows us the slope (3/2) and the y-intercept (5). We've successfully transformed the equation from point-slope form to slope-intercept form, giving us a complete picture of the line's characteristics. This process of simplification is a key skill in algebra and allows us to express linear equations in their most useful forms. In the next section, we'll summarize our findings and discuss the significance of the result.

Final Result and Conclusion

The equation of the line that passes through the point (2, 8) and has a slope of 3/2 is:

y = (3/2)x + 5

We found this equation by first using the point-slope form and then converting it to slope-intercept form. This process demonstrates a fundamental technique in algebra for determining the equation of a line given specific information. Understanding how to manipulate these equations is crucial for various mathematical and real-world applications.

In conclusion, finding the equation of a line given a point and a slope involves understanding the slope-intercept and point-slope forms, substituting the given values, and simplifying the equation. This skill is essential for anyone studying mathematics or working with linear relationships. By mastering these techniques, you can confidently tackle a wide range of problems involving lines and their equations.

For further exploration of linear equations and related concepts, you can visit resources like Khan Academy's Algebra I section.