Equation Of A Line: Slope 3 Through Point (-2, 4)

by Alex Johnson 50 views

Hey math enthusiasts! Let's dive into a fundamental concept in algebra: finding the equation of a line. Specifically, we're going to explore how to determine the equation of a line when we know its slope and a point it passes through. In this case, our line has a slope of 3 and goes through the point (-2, 4). Sounds straightforward, right? Well, it is! And we'll break it down step-by-step to make sure you grasp it completely. Understanding this is super important because it's a building block for so many other math concepts. Ready to get started?

Understanding the Basics: Slope and Point-Slope Form

First things first, let's refresh our memory on the key elements we'll be using. The slope of a line tells us how steep it is. It's the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. In our problem, the slope (m) is given as 3. This means that for every 1 unit we move to the right, the line goes up 3 units. Now, we'll introduce the point-slope form of a linear equation. This is a handy tool when we have the slope of a line and a point on that line. The point-slope form is written as: y - y₁ = m(x - x₁)

Where:

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

See? No sweat! The point-slope form is essentially a direct translation of the information we have: the slope and a point. We'll be using this formula to plug in the values we know (the slope and the coordinates of the point) to eventually get the equation in a form that's easier to interpret, such as the slope-intercept form.

Now, let's plug in the given values. Remember, our slope (m) is 3, and our point is (-2, 4). This means x₁ = -2 and y₁ = 4. Substituting these values into the point-slope form, we get: y - 4 = 3(x - (-2)).

Simplified to: y - 4 = 3(x + 2). And there you have it, the point-slope equation! Keep going, the hardest part is over!

Transforming to Slope-Intercept Form

Great! We now have the equation in point-slope form: y - 4 = 3(x + 2). However, the most common form for a linear equation is the slope-intercept form, which is written as: y = mx + b. In this form:

  • m represents the slope (which we already know is 3).
  • b represents the y-intercept (the point where the line crosses the y-axis).

Our mission now is to convert the point-slope form to the slope-intercept form. This involves a bit of algebra, but it’s really just a matter of simplifying and rearranging the equation. Let’s do it step by step:

  1. Distribute the slope: Multiply the 3 through the parentheses on the right side of the equation: y - 4 = 3x + 6.
  2. Isolate y: Add 4 to both sides of the equation to get y by itself: y = 3x + 6 + 4.
  3. Simplify: Combine the constants: y = 3x + 10.

And there we have it! The equation of the line in slope-intercept form is y = 3x + 10. From this form, we can easily see that the slope is 3 (as given) and the y-intercept is 10. This tells us the line crosses the y-axis at the point (0, 10). Pretty cool, right? This process, in essence, allows us to take a piece of information (the slope and a point) and translate it into a complete description of the line in a way that we can use it to determine points, graph, and much more.

This simple algebraic manipulation is key, not just for this problem, but for understanding and using linear equations in various contexts. Remember, these methods are not only useful for academic purposes, but also have real-world applications in fields like finance, engineering, and data analysis!

Visualizing the Line: Understanding the Graph

Okay, so we've found the equation y = 3x + 10. But what does this equation look like? Let's take a look at the graph. A quick way to visualize this is to consider the slope and the y-intercept. As we already know, the line has a slope of 3, meaning it goes up 3 units for every 1 unit it moves to the right. Also, the y-intercept is 10, meaning the line crosses the y-axis at the point (0, 10).

To graph this line, we can start by plotting the y-intercept (0, 10) on the coordinate plane. Then, we can use the slope to find another point. Since the slope is 3, start at the y-intercept and go up 3 units and to the right 1 unit. This gives us another point on the line. We can do this multiple times. Connecting these two points gives us the line representing the equation y = 3x + 10.

Alternatively, we can use the point we were given, (-2, 4). Since this point lies on the line, we can plot it, then, use the slope to get another point. Starting from (-2,4), go up 3 units and right 1 unit to get to (-1, 7). Keep repeating until you have enough points to connect. Notice how the line passes through (-2, 4) as the question stated. It's like magic, but it’s just the power of equations! Drawing the graph helps us to understand how changes in x affect changes in y. It brings the abstract concept of an equation to life, making it visually understandable.

By graphing the line, we can also see its relationship to the x-axis and the y-axis, and can further understand the direction in which the line will extend indefinitely. Using the knowledge of the slope allows you to determine if the line goes up to the right or down to the right. Visualizing the line is very important in understanding all the information you have calculated so far.

Putting It All Together: A Summary

Let's recap what we've accomplished:

  1. We were given the slope of a line (m = 3) and a point it passes through ((-2, 4)).
  2. We used the point-slope form: y - y₁ = m(x - x₁). Substituting our values gave us y - 4 = 3(x + 2).
  3. We converted this to the slope-intercept form: y = 3x + 10.
  4. We understood that the slope (3) and y-intercept (10) allowed us to graph the line.

And just like that, you've successfully found the equation of a line given its slope and a point! You can use this knowledge to solve a variety of problems in algebra and beyond. This is the foundation upon which more advanced mathematical concepts are built, and understanding the core elements such as the slope, the point, and the various forms of a linear equation, will prove to be very useful in many areas of mathematics. The ability to manipulate and convert equations from one form to another is a powerful tool to have. This is a skill that will certainly make solving equations, graphing, and analyzing linear relationships easier.

This method can be expanded on. If we only have two points, we can first calculate the slope, then use the methods described above to find the equation. In more advanced mathematics, such as calculus, these methods are used for finding tangents to curves, and in physics, they can be used to describe the motion of objects. So, congratulations, you have learned a very important mathematical concept!

Conclusion: Your Next Steps

Keep practicing! The more you work with linear equations, the more comfortable you'll become. Try different problems with different slopes and points, and see if you can graph the lines accurately. Experiment with different forms of the equation and understand how they relate to each other. Challenge yourself by working backwards: given the equation, can you find the slope and a point on the line? It is also useful to review what happens when lines are parallel or perpendicular, and see how these relate to the slope.

Remember, math is all about practice and persistence. Don't be afraid to make mistakes—they're opportunities to learn! With each problem you solve, you'll build a stronger understanding of the concepts and gain confidence in your abilities. Happy calculating!

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