Question Video: Finding the Equation of a Straight Line Mathematics • Third Year of Preparatory School

Video Transcript

Determine the equation of the line which passes through point negative seven, negative eight and it’s perpendicular to the line meeting the 𝑥-axis at nine and the 𝑦-axis at one.

The first word we’re gonna look at is perpendicular cause we’re told that the line that we need to find the equation of and the other line are perpendicular to one another. But how does that help us? Well it helps us because actually if we have the slope of two lines that perpendicular to each other, then if you multiply their slopes together you get an answer of negative one. So we can actually use that to find the slope of one line if we know the slope of the other.

Okay, great! So now we’ll need to find the slope of one of our lines. Well first of all, we’ve actually got two points that we can write down from the information we’ve got, cause we know that the line meets the 𝑥-axis at nine and the 𝑦-axis at one. So therefore, we know that there are two points, the first point being nine, zero. And that’s cause it meets the 𝑥-axis at nine. So that means the 𝑥-coordinate is gonna be nine. before I actually cross into the 𝑥-axis, then the 𝑦 is gonna be zero. And if we meet the 𝑦-axis at one, then it means the 𝑥-coordinate at that point is gonna be zero. So it’s gonna be zero, one.

Okay, great! So we’ve got two points, but how are these helpful? Well, we actually have a formula to help us find out the slope. And that is that 𝑚, our slope, is equal to 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. And what this actually means is the change in 𝑦 divided by the change in 𝑥. So when we have two points, we see the change in 𝑦 between those points and then we divided by the change in 𝑥 between those points, okay? So let’s do that using the two points we have.

So what I’ve done is I’ve actually labeled our coordinates to help us. So we’ve got 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two. And then what I’m gonna do is actually call our slope 𝑚 one. So to find 𝑚 one, we’re gonna now use the formula with the values that we’ve got. Therefore, we can say that 𝑚 one is gonna be equal to one minus zero, cause that’s our 𝑦 two minus our 𝑦 one, over zero minus nine, our 𝑥 two minus our 𝑥 one, which should give us a slope of negative one over nine.

Okay, great! So we’ve now found the slope of the other line. So what we want to see now is find the slope of the line that we want to find the equation of. And we can do that using our rule that we know for perpendicular lines cause we know that 𝑚 one multiplied by 𝑚 two is gonna be equal to negative one. So therefore, negative one over nine multiplied by 𝑚 two is equal to negative one. So what we want to is actually divide both sides of the equation here by negative one over nine. So when we do that, we get 𝑚 two. So our slope of the line that we are looking for is equal to negative one over negative one over nine.

Alright, so how did we work that out? What does it mean? Well actually what it means is the negative reciprocal. So we can say that actually it’s gonna be the negative reciprocal of negative one over nine. So therefore, 𝑚 two is gonna be equal to nine. Well we get that because actually dividing by negative one over nine is the same as multiplying by negative nine over one. Well if you multiply negative one by negative nine over one, you just get nine cause negative multiplied by a negative is a positive.

Okay, great! So we’ve now found 𝑚 two. Well now what we want to do is actually determine the equation of the line. And we have a point that it goes through, which is negative seven and negative eight, and we have its slope, which is equal to nine. So therefore, we can actually do is use the slope-point equation, which is a general equation, to help us find the equation of the straight line. And this tell us that 𝑦 minus 𝑦 one is equal to 𝑚 multiplied by 𝑥 minus 𝑥 one, where 𝑦 one and 𝑥 one are the coordinates of our point.

Okay, so let’s use this to actually find the equation of our line. So what we’ve have done is actually substituted in our 𝑥- and 𝑦- values. So we have 𝑦 minus, and then we’ve got negative eight that’s because our 𝑦 coordinate was negative eight, then is equal to nine because that’s our slope multiplied by 𝑥 minus. Then we’ve got negative seven because that was our 𝑥-coordinate. Well then, it’s worth nothing at this point we’ve actually got 𝑦 minus negative eight and 𝑥 minus negative seven. And if you actually subtract the negative then it turns into an add, so actually 𝑦 add eight. And we’ve got 𝑥 add seven.

So next what I’m gonna do is actually expand the parentheses. So when I do that, I get 𝑦 plus eight is equal to nine 𝑥 cause nine multiplied by 𝑥 gives us nine 𝑥 and then add 63, and that’s because nine multiplied by positive seven gives us positive 63. So now what I’m gonna do is actually to subtract eight from each side of the equation. And when we do that, we get 𝑦 is equal to nine 𝑥 plus 55.

So this is one way that we can leave the equation. So we can leave it in the form of 𝑦 equal 𝑚𝑥 plus 𝑐. And the other 𝑦 we can actually leave it is as nine 𝑥 minus 𝑥 plus 55 is equal to zero. So we leave it equal zero we do that by subtract 𝑦 from each side of the equation. Okay, great! So we’ve now solved the problem, and we said that the equation of the line which passes through point negative seven negative eight and is perpendicular to the line meets the 𝑥-axis at nine and the 𝑦-axis at one is 𝑦 equals nine 𝑥 plus 55 or nine 𝑥 minus 𝑦 plus 55 equal zero.