Line Equation: Point (3,-4) & Slope -5/3 Explained

by Alex Johnson 51 views

Let's dive into the fascinating world of linear equations! Today, we're going to tackle a specific problem: finding the equation of a line that passes through the point (3, -4) and has a slope of -5/3. Don't worry if this sounds intimidating; we'll break it down step by step in a friendly and easy-to-understand way. By the end of this article, you'll not only know how to solve this particular problem but also grasp the underlying concepts so you can confidently tackle similar challenges in the future. So, grab your metaphorical pencils (or your actual ones, if you prefer), and let's get started!

Understanding the Basics of Linear Equations

Before we jump into the specific problem, let's make sure we're all on the same page with the fundamental concepts. A linear equation represents a straight line on a graph. The most common form of a linear equation is the slope-intercept form, which looks like this:

  • y = mx + b

Where:

  • y represents the vertical coordinate.
  • x represents the horizontal coordinate.
  • m represents the slope of the line.
  • b represents the y-intercept (the point where the line crosses the y-axis).

The slope (m) is a crucial concept. It tells us how steep the line 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 larger the absolute value of the slope, the steeper the line. In our case, we have a negative slope of -5/3, which means our line will be sloping downwards.

The y-intercept (b) is simply the value of y when x is 0. It's the point where the line intersects the vertical y-axis. Knowing the slope and the y-intercept gives us a complete picture of the line.

Now that we've refreshed our understanding of the slope-intercept form, let's introduce another useful form: the point-slope form. This form is particularly handy when we know a point on the line and the slope, which is exactly what we have in our problem!

Introducing the Point-Slope Form

The point-slope form of a linear equation is given by:

  • y - y1 = m(x - x1)

Where:

  • (x1, y1) represents a specific point on the line.
  • m represents the slope of the line.

Notice how this form directly incorporates a point on the line and the slope. This makes it incredibly convenient for situations like ours, where we're given a point and a slope. The beauty of the point-slope form is that it allows us to construct the equation of the line without explicitly needing the y-intercept. We can simply plug in the given values and then, if desired, convert the equation to slope-intercept form.

This form is derived directly from the definition of slope. Remember that slope is the change in y divided by the change in x (rise over run). If we consider two points on the line, (x, y) and (x1, y1), the slope m can be expressed as:

  • m = (y - y1) / (x - x1)

Multiplying both sides of this equation by (x - x1) gives us the point-slope form: y - y1 = m(x - x1). So, you see, it's all connected!

Now that we have the point-slope form in our toolkit, we're ready to tackle our main problem. Let's move on to applying this knowledge to find the equation of the line.

Applying the Point-Slope Form to Our Problem

Okay, let's get down to business! We're given the point (3, -4) and the slope -5/3. We want to find the equation of the line that passes through this point and has this slope. The point-slope form is our weapon of choice here:

  • y - y1 = m(x - x1)

Let's identify our values:

  • (x1, y1) = (3, -4)
  • m = -5/3

Now, we simply plug these values into the point-slope form:

  • y - (-4) = (-5/3)(x - 3)

Notice the careful use of parentheses, especially when dealing with negative numbers. This is crucial to avoid errors in our calculations. Simplifying the equation, we get:

  • y + 4 = (-5/3)(x - 3)

We've now successfully applied the point-slope form and obtained the equation of the line in this form. However, it's often desirable to express the equation in slope-intercept form (y = mx + b). So, let's take the next step and convert our equation.

Converting to Slope-Intercept Form

To convert our equation from point-slope form to slope-intercept form, we need to isolate y on one side of the equation. This involves distributing the slope and then performing some algebraic manipulations. Our equation currently looks like this:

  • y + 4 = (-5/3)(x - 3)

First, let's distribute the -5/3 on the right side:

  • y + 4 = (-5/3)x + (-5/3)(-3)
  • y + 4 = (-5/3)x + 5

Now, to isolate y, we subtract 4 from both sides of the equation:

  • y + 4 - 4 = (-5/3)x + 5 - 4
  • y = (-5/3)x + 1

And there you have it! We've successfully converted the equation to slope-intercept form. We can now clearly see the slope (-5/3) and the y-intercept (1). This equation represents the line that passes through the point (3, -4) and has a slope of -5/3.

Verifying Our Solution

It's always a good idea to verify our solution to make sure we haven't made any mistakes. We can do this in a couple of ways:

  1. Plug in the point (3, -4) into our equation: If the equation holds true, then the point lies on the line.
  2. Check the slope: The coefficient of x in the slope-intercept form should match the given slope.

Let's start by plugging in the point (3, -4) into our equation, y = (-5/3)x + 1:

  • -4 = (-5/3)(3) + 1
  • -4 = -5 + 1
  • -4 = -4

The equation holds true! This confirms that the point (3, -4) lies on the line represented by our equation.

Next, let's check the slope. In our slope-intercept form, y = (-5/3)x + 1, the coefficient of x is -5/3, which matches the given slope. So, everything checks out!

We've successfully found the equation of the line and verified our solution. Give yourself a pat on the back! You've conquered this linear equation challenge.

Alternative Method: Direct Substitution into Slope-Intercept Form

While the point-slope form is a powerful tool, there's another way to approach this problem. We can directly substitute the given point and slope into the slope-intercept form (y = mx + b) and solve for the y-intercept (b). Let's see how this works.

We start with the slope-intercept form:

  • y = mx + b

We know the slope (m = -5/3) and a point ((x, y) = (3, -4)). Let's plug these values into the equation:

  • -4 = (-5/3)(3) + b

Now, we solve for b:

  • -4 = -5 + b
  • b = -4 + 5
  • b = 1

We've found that the y-intercept is 1. Now we can plug the slope (m = -5/3) and the y-intercept (b = 1) back into the slope-intercept form:

  • y = (-5/3)x + 1

This is the same equation we obtained using the point-slope form! This demonstrates that there are often multiple paths to the same solution in mathematics. Choosing the method that feels most comfortable and efficient for you is key.

Conclusion: Mastering Linear Equations

Congratulations! You've successfully navigated the process of finding the equation of a line given a point and a slope. We explored the point-slope form, converted to slope-intercept form, verified our solution, and even looked at an alternative method. You've not only solved a specific problem but also deepened your understanding of linear equations.

Remember, the key to mastering mathematics is practice and a willingness to break down problems into smaller, manageable steps. Keep exploring, keep questioning, and keep building your skills. Linear equations are a fundamental concept in mathematics and have wide-ranging applications in various fields, so the effort you put in now will pay off in the long run.

If you're eager to expand your knowledge further, I highly recommend checking out resources like Khan Academy's section on linear equations for more examples, practice problems, and in-depth explanations.

Happy learning, and keep those lines straight!