Finding Α For Perpendicular Lines: A Step-by-Step Solution

Hey guys! Today, we're diving into a fun algebra problem where we need to figure out when two lines are perpendicular. Specifically, we're looking for the value of α that makes the lines (α + 1)x + (3-α)y - 8 = 0 and (α - 3)x + (2α - 3)y = 0 perfectly perpendicular to each other. Sounds like a challenge? Don't worry, we'll break it down step by step so it's super easy to follow. Let's get started!

Understanding Perpendicular Lines

Before we jump into the algebra, let's quickly recap what it means for lines to be perpendicular. Perpendicular lines are lines that intersect at a right angle (90 degrees). A key concept here is the relationship between their slopes. If we have two lines, line 1 with slope m1 and line 2 with slope m2, they are perpendicular if and only if the product of their slopes is -1. In mathematical terms:

m1 * m2 = -1

This is our golden rule for this problem. We need to find the slopes of our given lines and then use this rule to solve for α. This relationship is crucial for solving this problem. Understanding this principle will help you tackle similar problems in the future. Remember, it's not just about memorizing a formula; it's about understanding why the formula works. When you grasp the underlying concept, you can apply it in various situations and even derive it yourself if you ever forget the exact formula. So, always focus on the 'why' behind the 'what'!

Finding the Slopes of the Lines

Okay, so we know that the product of the slopes of perpendicular lines is -1. But how do we find the slopes of the lines given in the problem? We have two equations:

1. (α + 1)x + (3 - α)y - 8 = 0
2. (α - 3)x + (2α - 3)y = 0

To find the slopes, we need to rewrite these equations in the slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. Let's tackle the first equation.

Rewriting the First Equation

We have (α + 1)x + (3 - α)y - 8 = 0. Our goal is to isolate y on one side of the equation. Here's how we do it:

1. Subtract (α + 1)x and add 8 to both sides: (3 - α)y = -(α + 1)x + 8
2. Divide both sides by (3 - α) to solve for y: y = [-(α + 1) / (3 - α)]x + [8 / (3 - α)]

Now we have the equation in slope-intercept form. The slope of the first line, which we'll call m1, is the coefficient of x:

m1 = -(α + 1) / (3 - α)

Rewriting the Second Equation

Next, we'll do the same thing for the second equation, (α - 3)x + (2α - 3)y = 0. Again, we want to isolate y:

1. Subtract (α - 3)x from both sides: (2α - 3)y = -(α - 3)x
2. Divide both sides by (2α - 3) to solve for y: y = [-(α - 3) / (2α - 3)]x

The slope of the second line, which we'll call m2, is the coefficient of x:

m2 = -(α - 3) / (2α - 3)

Great! We've found the slopes of both lines. Now we're ready to use our golden rule for perpendicular lines.

Applying the Perpendicularity Condition

Remember our rule? Two lines are perpendicular if the product of their slopes is -1. So, we have:

m1 * m2 = -1

Let's plug in the slopes we found:

[(-(α + 1) / (3 - α))] * [(-(α - 3) / (2α - 3))] = -1

This looks a bit intimidating, but don't worry, we'll simplify it. Notice that we have two negative signs in the numerators, which will cancel each other out. So, we can rewrite the equation as:

[(α + 1)(α - 3)] / [(3 - α)(2α - 3)] = -1

Now, let's get rid of the fraction by multiplying both sides by (3 - α)(2α - 3):

(α + 1)(α - 3) = -1 * (3 - α)(2α - 3)

Solving for α

Time to expand and simplify the equation. First, let's expand both sides:

Left side: (α + 1)(α - 3) = α² - 3α + α - 3 = α² - 2α - 3

Right side: -1 * (3 - α)(2α - 3) = -1 * (6α - 9 - 2α² + 3α) = -1 * (-2α² + 9α - 9) = 2α² - 9α + 9

Now we have:

α² - 2α - 3 = 2α² - 9α + 9

Let's move all the terms to one side to get a quadratic equation. Subtract α² from both sides, add 2α to both sides, and add 3 to both sides:

0 = α² - 7α + 12

Now we need to solve this quadratic equation. We can either use the quadratic formula or try to factor it. In this case, it's factorable. We're looking for two numbers that multiply to 12 and add up to -7. Those numbers are -3 and -4.

So, we can factor the equation as:

0 = (α - 3)(α - 4)

This gives us two possible solutions for α:

1. α - 3 = 0 => α = 3
2. α - 4 = 0 => α = 4

Checking for Extraneous Solutions

We have two potential solutions for α, but we need to be careful. Remember those denominators we had in our slope expressions? We can't have a denominator equal to zero, as that would make the slope undefined. Let's check our solutions.

Checking α = 3

If α = 3, then the denominator of m1 is (3 - α) = (3 - 3) = 0. Uh-oh! This means α = 3 makes the slope undefined, so it's an extraneous solution. We have to discard it.

Checking α = 4

If α = 4, then:

• The denominator of m1 is (3 - α) = (3 - 4) = -1 (okay)
• The denominator of m2 is (2α - 3) = (2 * 4 - 3) = 5 (okay)

So, α = 4 is a valid solution.

The Final Answer

After all that work, we've found our answer! The lines (α + 1)x + (3-α)y - 8 = 0 and (α - 3)x + (2α - 3)y = 0 are perpendicular when:

α = 4

Conclusion

Woohoo! We did it! We successfully found the value of α that makes the two lines perpendicular. This problem was a great exercise in applying the concept of perpendicular lines and their slopes. We also got some practice with rewriting equations in slope-intercept form and solving quadratic equations. Remember the key takeaways:

• Perpendicular lines have slopes that multiply to -1.
• Rewrite equations in slope-intercept form (y = mx + b) to easily identify the slope.
• Always check for extraneous solutions, especially when dealing with fractions.

I hope this step-by-step solution was helpful, guys! Keep practicing, and you'll become a master of algebra in no time. If you have any questions or want to tackle more problems like this, let me know in the comments. Happy solving!