Finding 'a' On A Line: Step-by-Step Solution

Let's dive into a fun math problem! We're going to find the value of 'a' when a point (a, -4) lies on a line that also passes through two other points: (2, -2) and (-6, 2). It might sound tricky, but don't worry, we'll break it down together, step by step. This is a classic coordinate geometry problem, and mastering it will boost your problem-solving skills. Understanding lines and their equations is fundamental in various fields, from engineering to computer graphics, so let's get started!

Understanding the Basics: Slope and the Equation of a Line

Before we jump into the solution, let's refresh some key concepts. The most important one here is the slope of a line. The slope, often denoted by 'm', tells us how steep the line is. Mathematically, it's the change in the y-coordinate divided by the change in the x-coordinate between any two points on the line. Think of it as "rise over run." The equation of a line is another critical concept. There are a few ways to represent it, but the slope-intercept form (y = mx + b) is particularly useful here, where 'm' is the slope and 'b' is the y-intercept (the point where the line crosses the y-axis). Knowing these basics will make the problem much easier to tackle. We'll use these concepts to find the equation of the line passing through our given points, and then we'll use that equation to find the value of 'a'. This combination of slope and equation is a powerful tool in coordinate geometry, and you'll find it used in many different types of problems.

Step 1: Calculate the Slope (m)

Our first step is to determine the slope (m) of the line that passes through the points (2, -2) and (-6, 2). Remember, the slope is calculated as the change in y divided by the change in x. So, let's use the formula:

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

Let's designate (2, -2) as (x1, y1) and (-6, 2) as (x2, y2). Now we can plug in the values:

m = (2 - (-2)) / (-6 - 2) m = (2 + 2) / (-8) m = 4 / -8 m = -1/2

So, the slope of our line is -1/2. This means that for every 2 units we move to the right on the line, we move 1 unit down. This negative slope indicates that the line is decreasing as we move from left to right. The calculated slope is a crucial piece of information because it will help us determine the equation of the line and subsequently find the value of 'a'. Understanding how to calculate the slope is a fundamental skill in coordinate geometry.

Step 2: Find the Equation of the Line

Now that we have the slope (m = -1/2), we can find the equation of the line. We'll use the slope-intercept form: y = mx + b. We already know 'm', and we can use one of the given points (let's use (2, -2)) to solve for 'b', the y-intercept. Plug in the values:

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

Add 1 to both sides to isolate 'b':

-2 + 1 = b b = -1

So, the y-intercept (b) is -1. Now we have both 'm' and 'b', so we can write the complete equation of the line:

y = (-1/2)x - 1

This equation represents the unique line that passes through the points (2, -2) and (-6, 2). The equation is our key to finding the value of 'a'. We'll use it in the next step to determine what 'a' must be if the point (a, -4) lies on this line. The ability to determine the equation of a line given two points is a powerful tool in algebra and geometry.

Step 3: Substitute (a, -4) into the Equation

We know that the point (a, -4) lies on the line, which means its coordinates must satisfy the equation we just found: y = (-1/2)x - 1. So, we can substitute x = a and y = -4 into the equation:

-4 = (-1/2)a - 1

Now, our goal is to solve for 'a'. To do this, let's first add 1 to both sides of the equation:

-4 + 1 = (-1/2)a - 1 + 1 -3 = (-1/2)a

Next, to get 'a' by itself, we need to multiply both sides of the equation by -2:

-3 * -2 = (-1/2)a * -2 6 = a

Therefore, the value of a is 6. This means the point (6, -4) lies on the same line as (2, -2) and (-6, 2). By substituting the coordinates into the line's equation, we've successfully found the unknown value. This process of substituting coordinates into an equation is a common technique in coordinate geometry and is useful for verifying if a point lies on a particular line or curve.

Conclusion: The Value of 'a' and the Power of Coordinate Geometry

And there you have it! We've successfully found that the value of 'a' is 6. This means the point (6, -4) sits snugly on the line that passes through (2, -2) and (-6, 2). We did it by first calculating the slope of the line using the two given points. Then, we used the slope and one of the points to find the equation of the line in slope-intercept form. Finally, we plugged in the coordinates of the point (a, -4) into the equation and solved for 'a'. This entire process showcases the power and elegance of coordinate geometry. By connecting algebra and geometry, we can solve problems that might seem complicated at first glance. Mastering these fundamental concepts opens doors to more advanced topics in mathematics and its applications in various fields. Remember, practice makes perfect, so try solving similar problems to solidify your understanding.

For further exploration of linear equations and coordinate geometry, check out resources like Khan Academy's Linear Equations and Graphs.