Parallel & Perpendicular Lines: Find The Equation!

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Alright, let's dive into some coordinate geometry, guys! We're going to tackle the challenge of finding equations for lines that are either parallel or perpendicular to a given line and pass through a specific point. Specifically, we will find the equation of a line passing through the point (5,-2) that is parallel to the line 5x+6y=7. Then we will find a second equation for a line passing through the point (5,-2) that is perpendicular to the line 5x+6y=7. This involves understanding slopes, slope-intercept form, and how parallel and perpendicular lines relate to each other. Buckle up, it’s going to be a fun ride!

Understanding the Basics

Before we jump into solving our problem, let’s quickly review some fundamental concepts that'll help us along the way. This includes slope-intercept form, parallel lines, and perpendicular lines. Think of this as our pre-flight check before takeoff!

Slope-Intercept Form

The slope-intercept form of a linear equation is a way of writing the equation of a line that makes it super easy to identify the slope and y-intercept. The form looks like this:

y = mx + b

Where:

  • y is the dependent variable (usually plotted on the vertical axis).
  • x is the independent variable (usually plotted on the horizontal axis).
  • m is the slope of the line, which tells you how steep the line is and whether it increases or decreases as you move from left to right. Specifically, it represents the “rise over run,” or the change in y for every unit change in x.
  • b is the y-intercept, which is the point where the line crosses the y-axis (i.e., the value of y when x is 0).

Why is this form so useful? Well, if you have an equation in this form, you can immediately see the slope and y-intercept without having to do any algebra. This makes it incredibly convenient for graphing lines and understanding their behavior. For example, if you have the equation y = 2x + 3, you know the slope is 2 and the y-intercept is 3. This means the line rises 2 units for every 1 unit you move to the right, and it crosses the y-axis at the point (0, 3).

Parallel Lines

Parallel lines are lines that run in the same direction and never intersect. The most important thing to remember about parallel lines is that they have the same slope. That's right, identical slopes! If two lines have the same slope but different y-intercepts, they will be parallel. Think of train tracks running side by side—they have the same steepness (slope) and never meet.

Mathematically, if one line has the equation y = m1x + b1 and another line has the equation y = m2x + b2, then these lines are parallel if and only if m1 = m2. In other words, the coefficients of x in both equations must be equal.

Perpendicular Lines

Perpendicular lines, on the other hand, intersect each other at a right angle (90 degrees). The relationship between their slopes is a bit more interesting. If one line has a slope of m, then a line perpendicular to it will have a slope of -1/m. This is known as the negative reciprocal. To find the slope of a perpendicular line, you flip the fraction (reciprocal) and change the sign (negative).

For instance, if a line has a slope of 2 (which can be written as 2/1), the slope of a line perpendicular to it would be -1/2. Similarly, if a line has a slope of -3/4, the slope of a perpendicular line would be 4/3.

In summary, if one line has the equation y = m1x + b1 and another line has the equation y = m2x + b2, then these lines are perpendicular if and only if m1 = -1/m2, which can also be written as m1 * m2 = -1. Understanding these slope relationships is crucial for solving geometric problems and understanding spatial relationships.

Finding the Parallel Line

Okay, now that we have refreshed our foundational knowledge, let’s find the equation of a line that passes through the point (5, -2) and is parallel to the line 5x + 6y = 7. Here's a step-by-step approach to nail this:

Step 1: Convert to Slope-Intercept Form

First, we need to convert the given equation, 5x + 6y = 7, into slope-intercept form (y = mx + b). This will allow us to easily identify the slope of the given line.

Subtract 5x from both sides of the equation:

6y = -5x + 7

Now, divide both sides by 6:

y = (-5/6)x + 7/6

From this, we can see that the slope of the given line is -5/6. Remember, parallel lines have the same slope. Therefore, the line we're trying to find will also have a slope of -5/6.

Step 2: Use the Point-Slope Form

Now that we know the slope (m = -5/6) and a point it passes through ((5, -2)), we can use the point-slope form of a linear equation to find the equation of the parallel line. The point-slope form is:

y - y1 = m(x - x1)

Where:

  • (x1, y1) is the given point.
  • m is the slope.

Plug in the values:

y - (-2) = (-5/6)(x - 5)

Simplify:

y + 2 = (-5/6)(x - 5)

Step 3: Convert to Slope-Intercept Form (Again!)

To make the equation look neat and tidy in slope-intercept form, we need to distribute and isolate y:

y + 2 = (-5/6)x + 25/6

Subtract 2 from both sides (remember that 2 is the same as 12/6):

y = (-5/6)x + 25/6 - 12/6

Simplify:

y = (-5/6)x + 13/6

So, the equation of the line that passes through the point (5, -2) and is parallel to the line 5x + 6y = 7 is y = (-5/6)x + 13/6.

Finding the Perpendicular Line

Next up, let’s find the equation of a line that passes through the same point (5, -2) but is perpendicular to the line 5x + 6y = 7. Remember that the slopes of perpendicular lines are negative reciprocals of each other.

Step 1: Determine the Perpendicular Slope

We already found that the slope of the given line is -5/6. To find the slope of a line perpendicular to it, we need to take the negative reciprocal of -5/6. This means flipping the fraction and changing the sign:

Perpendicular slope = -1 / (-5/6) = 6/5

So, the slope of the perpendicular line is 6/5.

Step 2: Use the Point-Slope Form (Again!)

Now that we know the slope of the perpendicular line (m = 6/5) and the point it passes through ((5, -2)), we can use the point-slope form again:

y - y1 = m(x - x1)

Plug in the values:

y - (-2) = (6/5)(x - 5)

Simplify:

y + 2 = (6/5)(x - 5)

Step 3: Convert to Slope-Intercept Form (One Last Time!)

Distribute and isolate y to get the equation in slope-intercept form:

y + 2 = (6/5)x - 30/5

Subtract 2 from both sides (remember that 2 is the same as 10/5):

y = (6/5)x - 30/5 - 10/5

Simplify:

y = (6/5)x - 40/5

y = (6/5)x - 8

Therefore, the equation of the line that passes through the point (5, -2) and is perpendicular to the line 5x + 6y = 7 is y = (6/5)x - 8.

Summary

To wrap things up, we found two equations:

  • Parallel Line: y = (-5/6)x + 13/6
  • Perpendicular Line: y = (6/5)x - 8

We successfully navigated through the concepts of slope-intercept form, parallel lines, and perpendicular lines. Great job, guys! Whether you’re studying for a test or just expanding your mathematical horizons, understanding these relationships is super valuable. Keep practicing, and you’ll become a coordinate geometry pro in no time!