Equation Of Line JK In Standard Form A Coordinate Geometry Problem

Hey guys! Today, we're diving into a classic coordinate geometry problem. We've got a line, aptly named JK, cruising through two points: J at (-3, 11) and K at (1, -3). Our mission, should we choose to accept it (and we totally do!), is to find the equation of this line in standard form. Buckle up, because we're about to embark on a mathematical adventure!

Understanding the Problem

Before we jump into calculations, let's make sure we understand what the question is asking. We're given two points on a line and need to determine the equation of that line. The key here is the standard form of a linear equation. Remember that standard form looks like this:

$Ax + By = C$

Where A, B, and C are integers, and A is usually a positive integer. This form is super useful for various things, like quickly finding intercepts or comparing different lines. So, our goal is to massage our initial information (the two points) into this beautiful standard form equation.

Now, how do we get there? The journey involves a few key steps:

1. Calculate the slope (m): The slope tells us how steep the line is and its direction. We can find it using the formula:

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

   Where (x1, y1) and (x2, y2) are our given points.

2. Use the point-slope form: Once we have the slope, we can plug it into the point-slope form of a linear equation:

   $y - y1 = m(x - x1)$

   This form is excellent because it directly uses a point on the line and the slope.

3. Convert to standard form: Finally, we'll rearrange the equation from point-slope form into the coveted standard form (Ax + By = C). This usually involves a bit of algebraic manipulation – distributing, adding, and subtracting terms until we get the desired format.

So, are you ready to put on your math hats and get started? Let's break it down step by step.

Step 1: Calculating the Slope

First things first, let's find the slope (m) of line JK. We have the points J(-3, 11) and K(1, -3). Let's label them:

• J(-3, 11) as (x1, y1)
• K(1, -3) as (x2, y2)

Now, we plug these values into the slope formula:

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

$m = (-3 - 11) / (1 - (-3))$

$m = (-14) / (1 + 3)$

$m = -14 / 4$

We can simplify this fraction by dividing both the numerator and the denominator by 2:

$m = -7 / 2$

Alright! We've got our slope. The slope of line JK is -7/2. This tells us that for every 2 units we move to the right along the line, we move 7 units down. Now that we have the slope, we're one step closer to finding the equation in standard form.

Step 2: Using the Point-Slope Form

Now that we've calculated the slope ($m = -7/2$), we can use the point-slope form of a linear equation to express the line's equation. The point-slope form is:

$y - y1 = m(x - x1)$

We have the slope, m, and we have two points, J(-3, 11) and K(1, -3), that we can use as (x1, y1). It doesn't matter which point we choose; we'll get the same equation in the end. For the sake of demonstration, let's use point J(-3, 11). So, x1 = -3 and y1 = 11.

Plugging these values into the point-slope form, we get:

$y - 11 = (-7/2)(x - (-3))$

Simplifying the expression inside the parentheses:

$y - 11 = (-7/2)(x + 3)$

This is the equation of line JK in point-slope form. It's a perfectly valid way to represent the line, but remember, our ultimate goal is to get it into standard form (Ax + By = C). So, let's move on to the final step.

Step 3: Converting to Standard Form

We've got the equation in point-slope form:

$y - 11 = (-7/2)(x + 3)$

Now, let's transform it into standard form (Ax + By = C). This involves a bit of algebraic maneuvering. Here's how we'll do it:

1. Distribute the -7/2: Multiply -7/2 by both terms inside the parentheses:

   $y - 11 = (-7/2)x - (7/2)(3)$

   $y - 11 = (-7/2)x - 21/2$

2. Get rid of the fraction: To eliminate the fractions, we can multiply both sides of the equation by the least common denominator (LCD), which in this case is 2:

   $2(y - 11) = 2((-7/2)x - 21/2)$

   $2y - 22 = -7x - 21$

3. Rearrange the terms: Now, we want to get the x and y terms on the left side and the constant term on the right side. Add 7x to both sides:

   $7x + 2y - 22 = -21$

   Then, add 22 to both sides:

   $7x + 2y = -21 + 22$

   $7x + 2y = 1$

And there you have it! We've successfully transformed the equation into standard form:

$7x + 2y = 1$

This matches option B in the given choices.

Conclusion

So, after navigating through slopes, point-slope form, and some algebraic gymnastics, we've found the equation of line JK in standard form. The answer is B. 7x + 2y = 1.

Remember, guys, the key to solving these types of problems is to break them down into manageable steps. Understanding the underlying concepts, like the slope formula and the different forms of linear equations, is crucial. With a little practice, you'll be able to tackle any coordinate geometry challenge that comes your way!

Now, wasn't that a fun math adventure? Keep practicing, and you'll become a math whiz in no time!