Sovi AI

Slope of the perpendicular line: \(-\frac{3}{2}\)
y - intercept: \(-13\)
Function: \(y =-\frac{3}{2}x-13\)

Turn 2 Answer

Absolutely, let's break this down super slowly—no rush at all. Let's start with the basics first, okay?

First, when they give us a line like \( y = \frac{2}{3}x + 10 \), that's already in slope-intercept form (\( y = mx + b \)). The \( m \) here is the slope, which is just the "steepness" of the line. So in this equation, the slope (\( m \)) is \( \frac{2}{3} \). That's step 1—we're just pulling the slope right out of the given equation because it's already in the easy-to-read form.

Now, perpendicular lines. Think about perpendicular lines like the corner of a square—they cross at a 90-degree angle. The rule for their slopes is that they're negative reciprocals. A reciprocal is just flipping the fraction, and negative means we flip the sign too. So if the original slope is \( \frac{2}{3} \), the reciprocal is \( \frac{3}{2} \), and then we make it negative: \( -\frac{3}{2} \). That's how we get \( m_{\perp} = -\frac{3}{2} \)—that's the slope of the line we need to find.

Next, we have a point that our line has to go through: (-12, 5). We use the point-slope formula here, which is \( y - y_1 = m(x - x_1) \). The \( x_1 \) and \( y_1 \) are just the x and y values from our point. So \( x_1 = -12 \) and \( y_1 = 5 \), and our \( m \) is the perpendicular slope we just found, \( -\frac{3}{2} \).

Plugging those into the formula: \( y - 5 = -\frac{3}{2}(x - (-12)) \). Wait, \( x - (-12) \) is the same as \( x + 12 \), right? Because subtracting a negative is like adding a positive. So that simplifies to \( y - 5 = -\frac{3}{2}(x + 12) \).

Now we need to turn this into slope-intercept form (\( y = mx + b \)) so we can find the y-intercept. Let's distribute the \( -\frac{3}{2} \) on the right side. That means we multiply \( -\frac{3}{2} \) by x and by 12.

\( -\frac{3}{2} \times x = -\frac{3}{2}x \). Then \( -\frac{3}{2} \times 12 = -18 \) (because 12 divided by 2 is 6, and 6 times -3 is -18). So now our equation is \( y - 5 = -\frac{3}{2}x - 18 \).

Last step: we need to get y by itself. We add 5 to both sides to cancel out the -5 on the left. So \( y = -\frac{3}{2}x - 18 + 5 \). Then -18 + 5 is -13. So our final equation is \( y = -\frac{3}{2}x - 13 \). The y-intercept is the constant term at the end, which is -13.

Does any part of that feel fuzzy? We can stop and rehash any step you want—no questions are too small! 😊

Turn 3 Answer

Oh, great catch! That’s just two different ways to write the exact same point-slope formula—they’re totally interchangeable, I should’ve pointed that out earlier!

Let me show you: the version on your sheet is \( y = m(x - x_1) + y_1 \), and the one I used was \( y - y_1 = m(x - x_1) \). If you take the second one and add \( y_1 \) to both sides, you get the first one! So they’re the same equation, just rearranged.

And you’re 100% right—using your sheet’s formula, it would be \( y = -\frac{3}{2}(x - (-12)) + 5 \). Let’s solve it that way to prove it’s the same answer:

First, simplify \( x - (-12) \) to \( x + 12 \), so it becomes \( y = -\frac{3}{2}(x + 12) + 5 \).
Distribute the \( -\frac{3}{2} \): \( y = -\frac{3}{2}x - 18 + 5 \).
Then add -18 + 5: \( y = -\frac{3}{2}x - 13 \). Same exact result!

The reason I used the other version is just habit, but your sheet’s formula is totally valid (and maybe easier to follow if that’s what you’re learning right now). Does that clear up why the 5 was in different spots?

Turn 4 Answer

Exactly! It's just a little algebraic rearranging—no fancy tricks, just moving terms around to get y by itself.

Like, if you start with \( y - y_1 = m(x - x_1) \), adding \( y_1 \) to both sides cancels it out on the left, leaving you with \( y = m(x - x_1) + y_1 \). Same equation, just written to have y already isolated from the start.

It’s nice that there’s more than one way to write it—whichever feels less confusing to you is the one to use! Does that make total sense now, or is there still a part that feels wobbly?

Turn 5 Answer

Got it! Let's start fresh with your sheet's formula, since that's what you're working with, and take it step by step super clearly:

1. Grab the slope of the original line

The given line is \( y = \frac{2}{3}x + 10 \), which is in slope-intercept form (\( y = mx + b \)). So the slope (\( m \)) of this line is \( \frac{2}{3} \).

1. Find the slope of the perpendicular line

Perpendicular slopes are negative reciprocals (flip the fraction and switch the sign). So flip \( \frac{2}{3} \) to get \( \frac{3}{2} \), then make it negative: \( m_{\perp} = -\frac{3}{2} \). This is the slope of the line we need to find.

1. Plug into your sheet's point-slope formula

The formula is \( y = m(x - x_1) + y_1 \), where:

• \( m = -\frac{3}{2} \) (our perpendicular slope)
• \( x_1 = -12 \) (the x-value from the given point (-12, 5))
• \( y_1 = 5 \) (the y-value from the given point (-12, 5))

Plugging in: \( y = -\frac{3}{2}(x - (-12)) + 5 \)

1. Simplify the inside of the parentheses first

\( x - (-12) = x + 12 \), so now we have: \( y = -\frac{3}{2}(x + 12) + 5 \)

1. Distribute the slope to the terms in the parentheses

Multiply \( -\frac{3}{2} \) by x, and \( -\frac{3}{2} \) by 12:
\( -\frac{3}{2} \times x = -\frac{3}{2}x \)
\( -\frac{3}{2} \times 12 = -18 \) (since 12 ÷ 2 = 6, 6 × -3 = -18)
Now the equation is: \( y = -\frac{3}{2}x - 18 + 5 \)

1. Combine the constant terms to get slope-intercept form

Add -18 and 5: \( -18 + 5 = -13 \)
Final equation: \( y = -\frac{3}{2}x - 13 \)

We can pause at any step to unpack it more—just say the word! 😊

Turn 6 Answer

Yay, you got it! That’s such a win 😊 Let’s clear up that last part about the perpendicular slope—totally makes sense to feel unsure about it at first.

Think of slopes as how "steep" a line is, and which direction it goes. The original line had a slope of 2/3, which means for every 3 units you move right along the x-axis, you move up 2 units on the y-axis (it’s a gentle upward slope).

Perpendicular lines cross at a perfect 90° angle, so their slopes have to be total opposites:

1. Flip the fraction (that’s the "reciprocal" part): 2/3 becomes 3/2
2. Flip the sign (positive to negative, or vice versa): 3/2 becomes -3/2

So that -3/2 slope means for every 2 units you move right, you move DOWN 3 units (it’s a steeper downward slope). If you drew both lines, they’d look like a perfect corner, like the sides of a square. That’s why we use that negative reciprocal rule—math’s way of making sure the lines are truly perpendicular.

And just to double-check your final answers:

• Slope of perpendicular line: -3/2 (that’s the m in your final function)
• Y-intercept: -13 (the standalone number at the end, no x attached)
• Function: y = -3/2x -13 (you just missed the x in your message, but I know you meant it!)

Does that make the perpendicular slope feel a little more concrete? We can even draw a mental picture of it if that helps!

Turn 7 Answer

Ugh, that’s such a real feeling—like half the battle is just decoding what the question is actually asking, right? It’s like math problems sometimes speak their own secret language.

I totally get breaking it brick by brick. Let’s try a little trick for unpacking questions like this: when you see a problem like “Find the slope-intercept form of the line that contains (-12, 5) and is perpendicular to y = 2/3x + 10,” try splitting it into tiny, actionable “tasks” instead of one big question:

1. First, what’s given? A point (-12,5) and a line to be perpendicular to.
2. What do I need to find first? The slope of the new line (since perpendicular lines have that special slope rule).
3. What formula do I need to use next? Point-slope (since I have a slope and a point).
4. What form do I need to end with? Slope-intercept (y = mx + b), so I have to rearrange the point-slope equation to get there.

It’s like turning a giant puzzle into smaller, easier pieces. Does that splitting-up trick sound like it could help? Or is there a specific part of “decoding” the question that feels extra tricky?

Turn 8 Answer

Aw, that makes my day to hear! 🥰 Math can feel so overwhelming when it feels like you're stuck alone, so I'm happy we're breaking this down together.

And hey, even if you hit another wall with this (or any other math stuff later), you know where to find me. We'll take it as slow as you need, brick by brick. Wanna take a quick break from math and chat about something totally random now? Or do you want to run through another practice problem to lock it in?

Turn 9 Answer

No worries at all—sounds like you’ve earned a low-key vibe. Go easy on yourself, especially with the lack of sleep. And yeah, absolutely, I’ll be here whenever you hit a snag later. You’ve got this! 😊