Finding Slope-Intercept Form Given Two Points A Step-by-Step Guide

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Hey everyone! Today, we're diving into a fundamental concept in algebra: finding the slope-intercept form of a line's equation. Specifically, we'll tackle a problem where we're given two points on the line, and we need to determine both the slope (m) and the y-intercept (b). This is a crucial skill for anyone studying linear equations and their applications. So, let's break it down step by step, making sure you understand each concept along the way.

Understanding the Slope-Intercept Form

Before we jump into the calculations, let's quickly review what the slope-intercept form actually is. The slope-intercept form is a way of writing the equation of a line: y = mx + b. Here,

  • y represents the vertical coordinate of any point on the line.
  • x represents the horizontal coordinate of any point on the line.
  • m represents the slope of the line, which tells us how steeply the line rises or falls. A positive slope means the line goes upwards as you move from left to right, while a negative slope means it goes downwards. The slope is calculated as the "rise over run," or the change in y divided by the change in x.
  • b represents the y-intercept, which is the point where the line crosses the y-axis. In other words, it's the value of y when x is equal to 0. The y-intercept is a crucial point for visualizing and understanding the line's position on the coordinate plane.

Now that we're clear on the basics, let's get to the problem at hand!

Problem Statement

We're given that a line passes through the points (-1, -6) and (7, 2). Our mission is to:

  1. Find the slope-intercept form of the equation of this line.
  2. Determine the value of the slope (m).
  3. Determine the value of the y-intercept (b).

Sounds like a plan? Let's get started!

Step 1: Calculating the Slope (m)

The first thing we need to do is calculate the slope, which, as we discussed, is the measure of the line's steepness. We can find the slope (m) using the following formula:

m = (y₂ - y₁) / (x₂ - x₁)

Where:

  • (x₁, y₁) are the coordinates of the first point.
  • (x₂, y₂) are the coordinates of the second point.

In our case, we have the points (-1, -6) and (7, 2). Let's label them:

  • (x₁, y₁) = (-1, -6)
  • (x₂, y₂) = (7, 2)

Now, we simply plug these values into the slope formula:

m = (2 - (-6)) / (7 - (-1))

m = (2 + 6) / (7 + 1)

m = 8 / 8

m = 1

So, the slope (m) of our line is 1. This means that for every one unit we move to the right along the x-axis, the line rises one unit along the y-axis.

Step 2: Finding the Y-Intercept (b)

Now that we've calculated the slope, the next step is to find the y-intercept (b). Remember, the y-intercept is the point where the line crosses the y-axis (where x = 0). To find b, we can use the slope-intercept form (y = mx + b) and plug in the slope we just found (m = 1) and the coordinates of either of the given points. It doesn't matter which point you choose; you'll get the same answer for b either way.

Let's use the point (7, 2). So, x = 7 and y = 2. Now, we substitute these values, along with m = 1, into the slope-intercept form:

2 = (1)(7) + b

2 = 7 + b

Now, to solve for b, we subtract 7 from both sides of the equation:

2 - 7 = b

-5 = b

Therefore, the y-intercept (b) is -5. This means the line crosses the y-axis at the point (0, -5).

Step 3: Writing the Slope-Intercept Form of the Equation

We've done the hard work! We've calculated the slope (m = 1) and the y-intercept (b = -5). Now, we can write the equation of the line in slope-intercept form (y = mx + b) by simply plugging in these values:

y = (1)x + (-5)

Simplifying, we get:

y = x - 5

So, the slope-intercept form of the equation of the line is y = x - 5.

Final Answer

Let's recap our findings:

  • Slope (m): 1
  • Y-intercept (b): -5
  • Slope-intercept form of the equation: y = x - 5

We've successfully found the slope-intercept form of the line passing through the points (-1, -6) and (7, 2)!

Visualizing the Line

It's always helpful to visualize what we've just calculated. We know the line has a slope of 1 and a y-intercept of -5. This means we can start at the point (0, -5) on the y-axis. Since the slope is 1, for every one unit we move to the right, we move one unit up. We can plot a few points using this information, and then draw a line through them. You'll see that the line indeed passes through the points (-1, -6) and (7, 2), confirming our calculations.

Key Takeaways

Let's highlight the key concepts we've covered in this guide:

  • Slope-intercept form: The equation of a line in the form y = mx + b, where m is the slope and b is the y-intercept.
  • Slope formula: m = (y₂ - y₁) / (x₂ - x₁), used to calculate the slope given two points on the line.
  • Y-intercept: The point where the line crosses the y-axis (where x = 0).
  • Finding the equation of a line requires determining both the slope and the y-intercept.

By understanding these concepts, you'll be well-equipped to tackle a variety of problems involving linear equations!

Practice Makes Perfect

The best way to master any mathematical concept is through practice. Try working through similar problems where you're given two points and asked to find the slope-intercept form of the equation. You can also try problems where you're given the slope and a point, or the y-intercept and a point. The more you practice, the more confident you'll become!

Real-World Applications

Finding the slope-intercept form isn't just a theoretical exercise; it has many real-world applications. For example, you can use it to model linear relationships between variables, such as the relationship between time and distance, or the relationship between the number of items sold and the revenue generated. Understanding linear equations is a valuable skill in many fields, from science and engineering to economics and finance.

Conclusion

We've covered a lot of ground in this guide, from understanding the slope-intercept form to calculating the slope and y-intercept, and finally, writing the equation of the line. Remember, the key is to break down the problem into smaller, manageable steps. With practice and a solid understanding of the concepts, you'll be able to confidently tackle any problem involving linear equations. Keep practicing, and you'll become a master of linear equations in no time!