Line Equation: Points (-6, 0) & (0, -24) | Step-by-Step

by ADMIN 56 views

Hey guys! Today, we're diving into a fundamental concept in mathematics: finding the equation of a line when you're given two points. Specifically, we're going to tackle the problem of finding the equation of the line that passes through the points (-6, 0) and (0, -24). Don't worry, it's not as scary as it sounds! We'll break it down step-by-step, making sure you understand each part of the process. So, let's get started and turn this mathematical challenge into a piece of cake!

Understanding the Basics: Slope-Intercept Form

Before we jump into the calculations, let's quickly review the slope-intercept form of a linear equation. This is the form we'll be aiming for, and it looks like this:

y = mx + b

Where:

  • y is the dependent variable (usually plotted on the vertical axis)
  • x is the independent variable (usually plotted on the horizontal axis)
  • m is the slope of the line (how steep it is)
  • b is the y-intercept (the point where the line crosses the y-axis)

Knowing this form is crucial because once we find the values of m (the slope) and b (the y-intercept), we can plug them into this equation and voilà, we have the equation of the line. So, let's move on to the first step: calculating the slope.

Step 1: Calculating the Slope (m)

The slope of a line tells us how much the line rises (or falls) for every unit it runs horizontally. We can calculate the slope using the following formula:

m = (y₂ - y₁) / (x₂ - x₁)

Where:

  • (x₁, y₁) and (x₂, y₂) are the coordinates of the two points the line passes through.

In our case, we have the points (-6, 0) and (0, -24). Let's label them:

  • (x₁, y₁) = (-6, 0)
  • (x₂, y₂) = (0, -24)

Now, let's plug these values into the slope formula:

m = (-24 - 0) / (0 - (-6)) m = -24 / 6 m = -4

So, the slope of our line is -4. This means that for every 1 unit we move to the right along the x-axis, the line goes down 4 units on the y-axis. Now that we've found the slope, let's move on to finding the y-intercept.

Importance of Understanding Slope

It's super important to grasp what the slope actually represents. A positive slope means the line goes upwards as you move from left to right, while a negative slope (like ours) means the line goes downwards. A slope of 0 indicates a horizontal line, and an undefined slope indicates a vertical line. Understanding this concept will help you visualize the line and check if your calculated slope makes sense in the context of the given points. For instance, with the points (-6, 0) and (0, -24), we can visualize a line going downwards from left to right, which aligns perfectly with our negative slope of -4. This is a great way to ensure your calculations are on the right track!

Step 2: Finding the Y-Intercept (b)

The y-intercept is the point where the line crosses the y-axis. This is the point where x = 0. Lucky for us, one of the points we were given is (0, -24). This point has an x-coordinate of 0, which means it lies on the y-axis. Therefore, the y-intercept is -24.

So, we have:

b = -24

If you weren't given the y-intercept directly, you could also find it by plugging the slope (m) and the coordinates of one of the points into the slope-intercept form (y = mx + b) and solving for b. We'll demonstrate this method later, just to be thorough and show you another way to tackle this.

Alternative Method for Finding the Y-Intercept

Let's say we didn't immediately recognize that (0, -24) gave us the y-intercept. No worries! We can still find 'b' by plugging in the slope we calculated (m = -4) and one of our points into the slope-intercept form (y = mx + b). Let's use the point (-6, 0):

0 = (-4)(-6) + b 0 = 24 + b b = -24

As you can see, we arrive at the same answer: b = -24. This alternative method is useful when you're not directly given a point with an x-coordinate of 0. It's always good to have multiple tools in your mathematical toolbox!

Step 3: Writing the Equation of the Line

Now that we have the slope (m = -4) and the y-intercept (b = -24), we can plug these values into the slope-intercept form (y = mx + b) to get the equation of the line:

y = -4x + (-24)

Simplifying this, we get:

y = -4x - 24

And there you have it! This is the equation of the line that passes through the points (-6, 0) and (0, -24).

Verifying the Equation

To make sure our equation is correct, we can plug the coordinates of the original points into the equation and see if they satisfy it. Let's try the point (-6, 0):

0 = -4(-6) - 24 0 = 24 - 24 0 = 0

The equation holds true for (-6, 0). Now let's try the point (0, -24):

-24 = -4(0) - 24 -24 = 0 - 24 -24 = -24

The equation also holds true for (0, -24). This gives us confidence that our equation is correct!

Alternative Forms of Linear Equations

While the slope-intercept form (y = mx + b) is super useful and commonly used, it's worth noting that there are other ways to represent a linear equation. Two other important forms are:

  1. Point-Slope Form: This form is particularly handy when you have a point and the slope, but you want to express the equation without explicitly finding the y-intercept. The point-slope form is:

    y - y₁ = m(x - x₁)

    Where (x₁, y₁) is a known point on the line and m is the slope.

  2. Standard Form: The standard form of a linear equation is:

    Ax + By = C

    Where A, B, and C are constants. This form is useful in various contexts, such as solving systems of linear equations.

Converting Between Forms

It's possible to convert between these different forms of linear equations. For example, you can easily convert from point-slope form to slope-intercept form by simplifying the equation and isolating y. Similarly, you can convert from slope-intercept form to standard form by rearranging the terms. Understanding these conversions allows you to work with linear equations in the most convenient form for a given problem.

Real-World Applications of Linear Equations

Linear equations aren't just abstract mathematical concepts; they have tons of real-world applications! Here are a few examples:

  • Calculating Costs: Imagine you're hiring a plumber who charges a flat fee plus an hourly rate. You can use a linear equation to model the total cost based on the number of hours worked.
  • Predicting Sales: Businesses use linear equations to analyze sales trends and make predictions about future sales based on past performance.
  • Modeling Motion: In physics, linear equations can be used to describe the motion of objects moving at a constant speed.
  • Data Analysis: Linear regression, a statistical technique, uses linear equations to model the relationship between two variables in a dataset.

These are just a few examples, but they illustrate how linear equations are a powerful tool for understanding and modeling the world around us.

Conclusion: Mastering Linear Equations

So, there you have it! We've successfully found the equation of the line that passes through the points (-6, 0) and (0, -24). We broke down the problem into manageable steps: calculating the slope, finding the y-intercept, and then plugging those values into the slope-intercept form. Remember, the key is to understand the concepts behind the formulas. Once you grasp the meaning of slope and y-intercept, finding the equation of a line becomes much easier. Keep practicing, and you'll be a pro in no time!

The equation of the line is y = -4x - 24.

I hope this guide has been helpful! If you have any questions, feel free to ask. Happy problem-solving, guys!