Find Perpendicular Line Equation Through A Point

Hey guys, ever stumbled upon a math problem that made you scratch your head? You know, the kind where you're given a line and a point, and you need to find the equation of another line that's perpendicular to the first one and also goes through that specific point? Don't sweat it! This is a super common concept in coordinate geometry, and once you break it down, it's totally manageable. We're talking about finding the equation of a line that is perpendicular to 6y - 3x - 2 = 0 and passes through the point (4, -3). Stick with me, and we'll conquer this together!

First things first, let's get our heads around what it means for two lines to be perpendicular. In the world of lines on a graph, perpendicularity means they intersect at a perfect 90-degree angle. Think of the corner of a square or the hands of a clock at 3:00 or 9:00. Mathematically, this special relationship translates into a cool property involving their slopes. If you have two lines, Line 1 with slope $m_1$ and Line 2 with slope $m_2$, and they are perpendicular, then the product of their slopes is -1. That is, $m_1 * m_2 = -1$. Alternatively, you can think of it as the slope of one line being the negative reciprocal of the other line's slope. So, if $m_1$ is your slope, then $m_2 = -1/m_1$. This little nugget of information is the golden ticket to solving our problem.

Now, let's tackle the given line: 6y - 3x - 2 = 0. To use our slope trick, we need to figure out its slope. The easiest way to do this is to rearrange the equation into the familiar slope-intercept form, which is y = mx + b. Here, 'm' represents the slope and 'b' is the y-intercept. Let's isolate 'y':

Start with: $6y - 3x - 2 = 0$

Add $3x$ to both sides: $6y - 2 = 3x$

Add 2 to both sides: $6y = 3x + 2$

Now, divide everything by 6 to get 'y' all by itself: $y = (3x + 2) / 6$

This can be split into: $y = (3/6)x + (2/6)$

Simplifying the fractions gives us: y = (1/2)x + (1/3).

Boom! We've found the slope of our original line. In this case, the slope, let's call it $m_1$, is 1/2.

Remember that 90-degree angle rule? Since our new line needs to be perpendicular to this one, its slope ($m_2$) must be the negative reciprocal of $m_1$. So, if $m_1 = 1/2$, then $m_2 = -1 / (1/2)$. Dividing by a fraction is the same as multiplying by its reciprocal, so $m_2 = -1 * (2/1) = -2$. Our perpendicular line has a slope of -2. Awesome!

Using the Point-Slope Form

Alright, we've got the slope of our target line ($m = -2$) and we know it passes through the point (4, -3). What's next? This is where the point-slope form of a linear equation comes in handy. The point-slope form is written as y - y1 = m(x - x1), where 'm' is the slope and $(x1, y1)$ are the coordinates of a point on the line. We have all the ingredients we need!

Let's plug in our values: $m = -2$, $x1 = 4$, and $y1 = -3$. Remember that $y1$ is -3, so when we substitute it into $y - y1$, it becomes $y - (-3)$, which simplifies to $y + 3$.

So, the equation becomes: $y - (-3) = -2(x - 4)$

$y + 3 = -2(x - 4)$

Now, let's simplify this equation. We can distribute the -2 on the right side:

$y + 3 = -2x + (-2 * -4)$

$y + 3 = -2x + 8$

Our final step is to get the equation into a standard form, like the slope-intercept form (y = mx + b) or the general form (Ax + By + C = 0). Let's aim for the slope-intercept form, as it's super common and easy to read.

To isolate 'y', we just need to subtract 3 from both sides of the equation:

$y = -2x + 8 - 3$

$y = -2x + 5$

And there you have it! The equation of the line that is perpendicular to $6y - 3x - 2 = 0$ and passes through the point $(4, -3)$ is y = -2x + 5. Pretty neat, right?

Recap and Key Takeaways

Let's quickly recap what we did, guys. We started with a given line equation, 6y - 3x - 2 = 0, and a point (4, -3). Our mission was to find the equation of a line perpendicular to the given line and passing through that point.

• Step 1: Find the slope of the given line. We rearranged $6y - 3x - 2 = 0$ into slope-intercept form, $y = (1/2)x + (1/3)$, to find its slope, $m_1 = 1/2$.
• Step 2: Determine the slope of the perpendicular line. Since perpendicular lines have slopes that are negative reciprocals of each other, we found the slope of our new line, $m_2$, to be $-1 / (1/2) = -2$.
• Step 3: Use the point-slope form. With the perpendicular slope ($m = -2$) and the given point $(x1, y1) = (4, -3)$, we used the point-slope formula $y - y1 = m(x - x1)$.
• Step 4: Simplify the equation. Plugging in our values gave us $y - (-3) = -2(x - 4)$, which we simplified to $y + 3 = -2x + 8$, and finally to y = -2x + 5.

This process is super useful for all sorts of geometry problems. Whether you're dealing with finding tangents to curves, analyzing geometric shapes, or just practicing your algebra skills, understanding how to find perpendicular lines is a fundamental building block. Remember the key relationship: the product of the slopes of perpendicular lines is -1. Master that, and you're well on your way to solving many more complex problems. Keep practicing, and don't be afraid to tackle those challenging questions! You've got this!