Then this time we want to talk about the perpendicular lines here. If under one graph, one plane graph, if two lines are perpendicular, then we should know the slope. So for example, this is the y equal to m2x plus b2.
and then this is y equal to m1 x plus b1. If they are perpendicular, then m1 times m2 must be equal to negative 1. So from here, let's solve for maybe one slope, maybe m2. Then you may divide by m1, both sides. 2 becomes negative 1 over m2 here. So from the original m1, whenever you find perfect negative slope m2, then it becomes opposite reciprocal.
That's how you remember that here. Okay? So then one example. I'm gonna give you y equal to 2x plus 5 And then we want to find the perpendicular line passing through. Okay, it's the same way like before, but not the parallel line anymore.
So you need to analyze negative reciprocal as a perpendicular slope here. So given m equal to 2, which means you can rewrite this. 2 over 1 here.
So we can find perpendicular slope opposite which is negative reciprocal so become 1 over 2. That will be our slope. Now you know one point here so you can use point-slope formula to complete the equation of the line. So y minus 0 equal to negative 1 over 2 x minus 4. Okay, then y equal to negative 1 over 2x plus 2 basically.
Okay, so that will be answer for this question here. Okay, then what about equation is given as the standard formula such as 3x plus 5y. equal to 10. Okay? Then same thing.
Now you need to solve for y so that we can find the slope. Then this becomes minus 3x on the both sides. So 5y equal to negative 3x plus 10 and divide by 5. So y equal to negative 3 over 5x plus 2 here.
Okay. Now, let's find perpendicular line to this one and pass it through 1 comma negative 7. Maybe 3 comma negative 7 here. Okay.
So, to make it easier, I'm going to switch the 3 here. Then, from the given slope, And you need to analyze the perpendicular slope. So, perpendicular slope must be equal to opposite, which means plus, risk precursor 5 over 3. And you know one point here.
Therefore, you can make y minus negative 7 equal to 5 over 3, x minus 3 here. Now, You distribute, then y plus 7 equal to 5 over 3x minus 5. And now, you subtract 7, then y equal to 5 over 3x minus 12. So, that should be the answer for this problem here. But...
Like before, when we talk about parallel lines, because given equation is the standard form. So, let's switch this to the standard form as well. So, the first thing what I want to do multiply denominator value which is 3 to everything here. Then, this becomes 3y equal to 5x minus 36. Now, I'm gonna move 5x. to the other side, then this becomes negative 5x plus 3y equal to negative 36. It is fine answer, but we prefer writing positive sign the front one, so you multiply negative one everything, then this becomes 5x minus 3y equal to 36. Okay.
So let me bring the original given equation which is 3x plus 5y equal to 10. Okay. Now compare this equation and this equation. Do you see something here? Yes.
You see that basically it was originally 3 on the x value, it moved to y coefficient, and then 5 go to x coefficient, and the sign got opposite here. So, it means if given equation looks like ax plus by equal to c, the perpendicular line equation can be set up. x minus ay equal to d something like that. Okay?
So, look at one question. The given equation is 4x minus 3y equal to negative 6. And then you want to find out perpendicular lines of this given line. and pass through 2, 1. Okay? Then, based on this fact, you can switch 3 goes to x value, 4 goes to y value, and this sign becomes opposite.
So, 3x plus 4y equals to this so far. Okay? And then you plug 2 into x, 1 into y, then this becomes 3 times 2 plus 4 times 1 equal to d, which means 6 plus 4 equal to 10. That will be our d value here.
Therefore, answer becomes 3x plus 4y equal to 10. That will be the answer for this problem.