Equation And Slope Of Line Passing Through (10, 9) And (5, 12)

Hey everyone! Today, we're diving into a classic math problem: finding the equation of a line that passes through two given points. Specifically, we're going to work with the points (10, 9) and (5, 12). We'll also figure out the slope of this line. So, grab your pencils and let's get started!

Understanding the Basics: Slope and the Point-Slope Form

First, let's quickly recap some fundamental concepts. The slope of a line tells us how steep it is and in what direction it's going. A positive slope means the line is going upwards as you move from left to right, while a negative slope means it's going downwards. The slope is calculated as the change in the y-coordinates divided by the change in the x-coordinates.

The point-slope form is the equation we use to express the equation of the line that uses a single point on the line and the slope of the line. This form is particularly useful when we know a point on the line and the slope, which is exactly the situation we have here. The point-slope form looks like this:

y - y₁ = m(x - x₁)

Where:

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

Calculating the Slope

The slope is often the first thing we calculate when dealing with lines and points. To calculate the slope (m) of a line passing through two points (x₁, y₁) and (x₂, y₂) we use the following formula:

m = (y₂ - y₁) / (x₂ - x₁)

In our case, we have (10, 9) as (x₁, y₁) and (5, 12) as (x₂, y₂). Let's plug these values into the formula:

m = (12 - 9) / (5 - 10)

m = 3 / -5

m = -3/5

So, the slope of the line passing through the points (10, 9) and (5, 12) is -3/5. This negative slope tells us that the line is decreasing or going downwards as we move from left to right on the graph. Keep this slope value handy, because we'll need it in the next step to determine the equation of the line. Guys, this is a crucial step, so make sure you understand how we arrived at this slope!

Finding the Equation of the Line: Point-Slope Form in Action

Now that we've successfully calculated the slope (m = -3/5), we can move on to finding the equation of the line. Remember the point-slope form? It's our trusty tool for this task: y - y₁ = m(x - x₁). We already know the slope (m), and we have two points to choose from: (10, 9) and (5, 12). For this example, let's use (10, 9) as our (x₁, y₁) but it does not matter if you use the other point, you will get the same result. The beauty of the point-slope form is that it works with any point on the line.

Let's substitute the values into the point-slope form:

y - 9 = -3/5 (x - 10)

This is the equation of the line in point-slope form. However, it's often preferred to express the equation in slope-intercept form (y = mx + b) or standard form (Ax + By = C). Let's convert our equation to slope-intercept form first. To do this, we need to distribute the -3/5 and then isolate y:

y - 9 = -3/5x + (-3/5)(-10)

y - 9 = -3/5x + 6

Now, add 9 to both sides to isolate y:

y = -3/5x + 6 + 9

y = -3/5x + 15

So, the equation of the line in slope-intercept form is y = -3/5x + 15. This form tells us that the y-intercept (the point where the line crosses the y-axis) is 15. Knowing both the slope and the y-intercept gives us a clear picture of the line's behavior. We could also convert this to standard form by adding (3/5)x to both sides and then, if desired, multiplying through by 5 to eliminate fractions:

(3/5)x + y = 15

Multiply both sides by 5:

3x + 5y = 75

Thus, the equation of the line in standard form is 3x + 5y = 75. Whether you prefer slope-intercept form or standard form, the important thing is that you understand how to manipulate the equation to get it into the desired format. Great job, guys! We're making excellent progress.

Visualizing the Line and its Properties

To truly understand the line we've described, it's helpful to visualize it. Imagine a graph with the x and y axes. Our line passes through the points (10, 9) and (5, 12). As we calculated, the slope is -3/5, which means that for every 5 units we move to the right along the x-axis, we move 3 units down along the y-axis. The y-intercept is 15, meaning the line crosses the y-axis at the point (0, 15).

If you were to plot these points and draw the line, you'd see a line sloping downwards from left to right. This visual representation can solidify your understanding of the slope and the equation we derived. In fact, sketching a quick graph is often a good strategy when you're working on problems like these. It helps you to check if your calculations make sense. For example, if you calculated a positive slope but the line clearly slopes downwards, you'd know there was a mistake somewhere. Visualizing mathematical concepts is a powerful tool, so don't hesitate to use it! Now, let's recap what we've learned and think about how we can apply these skills to other problems.

Key Takeaways and Applications

So, what have we accomplished today? We successfully found the equation of the line passing through the points (10, 9) and (5, 12), and we determined its slope. We started by calculating the slope using the formula m = (y₂ - y₁) / (x₂ - x₁). Then, we used the point-slope form y - y₁ = m(x - x₁) to find the equation of the line. Finally, we converted the equation to slope-intercept form (y = mx + b) and standard form (Ax + By = C).

These skills are fundamental in algebra and have wide-ranging applications. Understanding how to find the equation of a line is crucial for solving various problems in mathematics, physics, engineering, and even economics. For instance, you might use these concepts to model the relationship between two variables, predict future values, or optimize a system.

The ability to work with linear equations is a building block for more advanced mathematical concepts. As you continue your mathematical journey, you'll find that these skills come in handy time and time again. So, keep practicing, keep exploring, and don't be afraid to tackle challenging problems. You've got this!

Practice Problems and Further Exploration

To solidify your understanding, try working through some practice problems. Here are a few ideas:

1. Find the equation of the line passing through the points (2, -3) and (4, 1).
2. Determine the slope and y-intercept of the line with the equation 2x - 3y = 6.
3. A line has a slope of 2 and passes through the point (1, 4). Find its equation.

Also, consider exploring related topics, such as parallel and perpendicular lines. How do their slopes relate to each other? Can you write the equation of a line that is parallel or perpendicular to a given line and passes through a specific point?

By tackling these types of problems, you'll not only reinforce your understanding of linear equations but also develop your problem-solving skills. Remember, mathematics is a journey of discovery, so embrace the challenges and enjoy the process. You guys are doing awesome! Let’s keep up the great work and continue to explore the exciting world of mathematics together.

In conclusion, mastering the concepts of slope and linear equations opens doors to a deeper understanding of mathematics and its applications in various fields. By understanding how to calculate slope, utilize the point-slope form, and manipulate equations into different forms, you equip yourself with essential tools for problem-solving and critical thinking. Keep practicing, stay curious, and continue to explore the fascinating world of mathematics!