Equation Of A Line: M(-3, 5) And N(2, 0) - Find The Slope

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Hey guys! Today, we're diving into a fundamental concept in coordinate geometry: finding the equation of a line when you're given two points. Specifically, we'll tackle the problem of finding the equation of the line MN, where M is the point (-3, 5) and N is the point (2, 0). Don't worry, it's not as intimidating as it sounds! We'll break it down step-by-step, making sure you understand every part of the process. So, let's get started and make linear equations our friends!

Step 1: Mastering the Slope - The Key to Our Line

In tackling the challenge of finding the line equation passing through points M(-3, 5) and N(2, 0), the slope is our initial critical component. The slope (often denoted as m) is super important, acting as the compass guiding our line’s direction and steepness across the coordinate plane. Think of it as the line's personality! To calculate the slope, we use a nifty formula that considers the change in the y-coordinates (the rise) divided by the change in the x-coordinates (the run). This gives us a numerical value representing how much the line goes up or down for every unit it moves to the right. Understanding the slope is crucial because it's the foundation upon which we build the entire equation of the line. A positive slope indicates an upward trend, a negative slope signifies a downward trend, a zero slope represents a horizontal line, and an undefined slope points to a vertical line. So, let’s dive deep into calculating this slope for our specific points M and N, unlocking the first key piece of our linear equation puzzle. This foundational step ensures we accurately represent the line's inclination, setting the stage for the subsequent steps in defining its equation. The slope formula is defined as:

m = (y2 - y1) / (x2 - x1)

Where:

  • (x1, y1) are the coordinates of the first point (M in our case).
  • (x2, y2) are the coordinates of the second point (N in our case).

Let's plug in the coordinates of our points M(-3, 5) and N(2, 0) into the formula:

  • x1 = -3
  • y1 = 5
  • x2 = 2
  • y2 = 0

So, the slope m is calculated as:

m = (0 - 5) / (2 - (-3))
  = -5 / (2 + 3)
  = -5 / 5
  = -1

Therefore, the slope of the line MN is -1. This negative slope tells us that the line slopes downwards as we move from left to right. Now that we've nailed the slope, we're one big step closer to finding the equation of the line. Next, we'll use this slope along with one of our points to determine the y-intercept, which will complete our equation.

Step 2: Unveiling the Y-Intercept - Where Our Line Crosses

Having computed the slope m as -1, our next move in defining the equation of line MN is to pinpoint the y-intercept. Think of the y-intercept as the line's home on the y-axis – it's the point where the line intersects this vertical axis. Knowing the y-intercept is vital because it, along with the slope, allows us to write the equation of the line in slope-intercept form, which is a super handy and commonly used format. To find this y-intercept, we'll use the slope-intercept form of a linear equation, which looks like this:

y = mx + b

Where:

  • y is the y-coordinate
  • m is the slope (which we already found!)
  • x is the x-coordinate
  • b is the y-intercept (the value we're trying to find)

We already know m = -1. To find b, we can plug in the coordinates of either point M or point N into this equation. It doesn't matter which point we choose because both points lie on the line, so they'll both satisfy the equation. Let's use point N (2, 0) because it has a 0 as one of its coordinates, which will simplify our calculations. So, we have x = 2 and y = 0. Plugging these values into the slope-intercept form, we get:

0 = (-1)(2) + b
0 = -2 + b

Now, we can solve for b by adding 2 to both sides of the equation:

0 + 2 = -2 + b + 2
2 = b

So, the y-intercept (b) is 2. This means the line MN intersects the y-axis at the point (0, 2). We've now successfully found both the slope and the y-intercept. With these two pieces of information, we're ready to write the full equation of the line. This is where things really come together, allowing us to express the relationship between x and y for every point on the line.

Step 3: Constructing the Equation - Putting It All Together

Now for the exciting finale! With the slope (m = -1) and the y-intercept (b = 2) in our grasp, we're fully equipped to write the equation of line MN. We'll use the slope-intercept form again, as it neatly incorporates these two key pieces of information. Remember, the slope-intercept form is:

y = mx + b

We simply substitute the values we found for m and b into this equation. So, replacing m with -1 and b with 2, we get:

y = (-1)x + 2

This can be simplified to:

y = -x + 2

And that's it! We've successfully derived the equation of the line MN. This equation, y = -x + 2, tells us everything we need to know about the line. For any x-value, we can plug it into this equation to find the corresponding y-value, and vice versa. This equation not only represents the line graphically on a coordinate plane but also mathematically defines the relationship between the x and y coordinates of every point on the line. This final step is the culmination of our efforts, providing a concise and powerful representation of the line that passes through points M(-3, 5) and N(2, 0).

Alternative Forms: Point-Slope Form

While the slope-intercept form (y = mx + b) is super useful, there's another way to represent the equation of a line called the point-slope form. It's particularly handy when you know a point on the line and the slope, which we do in our case! The point-slope form looks like this:

y - y1 = m(x - x1)

Where:

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

We already know m = -1. We can use either point M or N for (x1, y1). Let's use point M (-3, 5). Plugging these values into the point-slope form, we get:

y - 5 = -1(x - (-3))
y - 5 = -1(x + 3)

This is a perfectly valid equation for the line MN in point-slope form. If we wanted to, we could distribute the -1 and add 5 to both sides to convert it back to slope-intercept form, but the point-slope form is often useful in its own right. It highlights a specific point on the line and the slope, giving us a slightly different perspective on the line's characteristics.

Wrapping Up: You've Cracked the Code!

So, there you have it! We've successfully navigated the process of finding the equation of a line given two points. We started by calculating the slope, then we found the y-intercept, and finally, we combined these elements to write the equation in slope-intercept form. We even explored the point-slope form as an alternative representation. Remember, the key is to break down the problem into smaller, manageable steps, and you'll be solving these types of problems like a pro in no time. Keep practicing, and you'll master the art of linear equations! Great job, guys!