Perpendicular Line Equation: Step-by-Step Solution
Hey guys! 👋 Today, we're diving into a classic math problem: finding the equation of a line that's perpendicular to another line and passes through a specific point. It might sound intimidating, but trust me, we'll break it down into easy-to-follow steps. We'll tackle the problem where we need to find the equation of a line that passes through the point A(7, -3) and is perpendicular to the line whose equation is 2x - 5y = 8. Let's get started!
Understanding the Basics: Slopes and Perpendicular Lines
Before we jump into the calculations, let's quickly review some fundamental concepts. The slope of a line is a measure of its steepness and direction. It tells us how much the line rises (or falls) for every unit change in the horizontal direction. We often represent the slope with the letter 'm'. A line with a positive slope rises from left to right, while a line with a negative slope falls from left to right. A horizontal line has a slope of 0, and a vertical line has an undefined slope.
Now, what about perpendicular lines? Two lines are perpendicular if they intersect at a right angle (90 degrees). There's a special relationship between their slopes: the slopes of perpendicular lines are negative reciprocals of each other. This means if one line has a slope of 'm', the slope of a line perpendicular to it will be '-1/m'. This negative reciprocal relationship is crucial for solving our problem.
To illustrate this, imagine a line with a slope of 2. A line perpendicular to it would have a slope of -1/2. See how we flipped the fraction and changed the sign? This is the essence of the negative reciprocal. Another example: if a line has a slope of -3/4, a perpendicular line would have a slope of 4/3. Keep this concept in mind as we move forward. Understanding this relationship between slopes is key to finding the equation of a perpendicular line. We'll be using it extensively in the following steps, so make sure you've got it down! We can also think about this geometrically: if you rotate a line by 90 degrees, you're essentially flipping its slope and inverting its direction (hence the negative sign).
Finding the Slope of the Given Line
Our first step is to determine the slope of the line given by the equation 2x - 5y = 8. To do this, we need to rewrite the equation in slope-intercept form, which is y = mx + b, where 'm' is the slope and 'b' is the y-intercept. This form makes it super easy to identify the slope, as it's simply the coefficient of the 'x' term.
Let's rearrange the equation 2x - 5y = 8. First, we'll subtract 2x from both sides: -5y = -2x + 8. Then, we'll divide both sides by -5 to isolate 'y': y = (2/5)x - 8/5. Now, we have the equation in slope-intercept form! We can clearly see that the slope of the given line is 2/5. This is a crucial piece of information because we'll use it to find the slope of the perpendicular line.
Remember the negative reciprocal relationship? Since the slope of our given line is 2/5, the slope of any line perpendicular to it will be -5/2. We flipped the fraction and changed the sign! This -5/2 is the 'm' we'll be using in the next step when we construct the equation of the perpendicular line. So, we've successfully found the slope of the perpendicular line, which is -5/2. This is a significant step forward in solving our problem! Keep this value handy, as we'll be plugging it into the point-slope form of a linear equation shortly. By isolating 'y', we were able to reveal the slope hidden within the original equation. This technique is a fundamental skill in linear algebra, and mastering it will help you tackle a wide range of problems.
Determining the Slope of the Perpendicular Line
Now that we know the slope of the given line is 2/5, we can find the slope of the line perpendicular to it. Remember, the slopes of perpendicular lines are negative reciprocals of each other. This means we need to flip the fraction and change the sign.
The negative reciprocal of 2/5 is -5/2. So, the slope of the line perpendicular to 2x - 5y = 8 is -5/2. Let's call this slope m_perp. We have m_perp = -5/2. This value is crucial because it tells us the steepness and direction of our new line, the one we're trying to find the equation for. Understanding the negative reciprocal relationship is the key to unlocking this part of the problem. It's a simple rule, but it has powerful implications in geometry and linear algebra. We now know the direction our line needs to travel, and this will help us in the next step when we use the point-slope form.
Using the Point-Slope Form
To find the equation of the perpendicular line, we'll use the point-slope form of a linear equation. This form is particularly useful when we know a point that the line passes through (x1, y1) and its slope 'm'. The point-slope form is given by: y - y1 = m(x - x1).
We know the perpendicular line passes through the point A(7, -3), so x1 = 7 and y1 = -3. We also know the slope of the perpendicular line is m_perp = -5/2. Now, we can plug these values into the point-slope form:
y - (-3) = (-5/2)(x - 7)
Simplifying, we get:
y + 3 = (-5/2)(x - 7)
This is the equation of the line in point-slope form. While it's a perfectly valid equation, we usually want to express it in slope-intercept form (y = mx + b) or standard form (Ax + By = C). Let's convert it to slope-intercept form in the next step. The point-slope form is a powerful tool because it allows us to construct the equation of a line using minimal information. It bridges the gap between knowing a single point and knowing the entire line. By plugging in our values, we've created a mathematical representation of our perpendicular line, and we're just a few steps away from having it in a more familiar format.
Converting to Slope-Intercept Form
Now that we have the equation in point-slope form (y + 3 = (-5/2)(x - 7)), let's convert it to slope-intercept form (y = mx + b). This form makes it easy to visualize the line and identify its y-intercept.
To convert, we need to distribute the -5/2 on the right side of the equation and then isolate 'y'.
First, distribute the -5/2: y + 3 = (-5/2)x + 35/2
Next, subtract 3 from both sides: y = (-5/2)x + 35/2 - 3
To subtract 3, we need to express it as a fraction with a denominator of 2: 3 = 6/2
So, we have: y = (-5/2)x + 35/2 - 6/2
Combining the fractions, we get: y = (-5/2)x + 29/2
This is the equation of the perpendicular line in slope-intercept form! We can see that the slope is -5/2 (as we expected) and the y-intercept is 29/2. Converting to slope-intercept form gives us a clear picture of the line's behavior. We know how steeply it slopes and where it crosses the y-axis. This form is incredibly useful for graphing the line and comparing it to other lines. We're almost there! We have the equation in a commonly used form, and we're just one step away from potentially putting it in standard form as well.
Expressing the Equation in Standard Form (Optional)
While the slope-intercept form is great, sometimes we need the equation in standard form, which is Ax + By = C, where A, B, and C are integers, and A is usually positive. Let's convert our equation from slope-intercept form (y = (-5/2)x + 29/2) to standard form.
First, we want to get rid of the fractions. To do this, we can multiply both sides of the equation by 2:
2y = -5x + 29
Next, we want to move the 'x' term to the left side of the equation. Add 5x to both sides:
5x + 2y = 29
Now, we have the equation in standard form! A = 5, B = 2, and C = 29. Expressing the equation in standard form provides a clean and organized representation of the line. It's particularly useful when dealing with systems of linear equations. We've now successfully converted the equation to standard form, giving us a complete solution to the problem. We started with a point and a condition of perpendicularity, and we've arrived at the equation of the line that satisfies those requirements.
Final Answer
Therefore, the equation of the line that passes through A(7, -3) and is perpendicular to the line 2x - 5y = 8 is:
- Slope-intercept form: y = (-5/2)x + 29/2
- Standard form: 5x + 2y = 29
We did it, guys! We successfully found the equation of the perpendicular line using our knowledge of slopes, the point-slope form, and a little algebraic manipulation. Remember, practice makes perfect, so keep tackling these problems, and you'll become a pro in no time!
