Question Video: Finding the Equation of a Straight Line given Its π‘₯- and 𝑦-Intercepts Mathematics

What is the equation of the line with π‘₯-intercept βˆ’3 and 𝑦-intercept 4?

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Video Transcript

What is the equation of the line with π‘₯-intercept negative three and 𝑦-intercept four?

So the first thing we’re gonna do in this question is sketch our line. So what we’ve got first of all is an π‘₯-intercept of negative three, which means we know that our line crosses the π‘₯-axis at negative three. And we have a 𝑦-intercept at four. So it crosses our 𝑦-axis at four. So now if we join these points, we have our straight line. So what we want to do now is find its equation. And the form in which we’re gonna find the equation first of all is the point-slope form, which is 𝑦 minus 𝑦 sub one equals π‘š multiplied by π‘₯ minus π‘₯ sub one. And the reason we’re going to do this is because it’s the easiest way to tackle the problem because what we can do is identify specific points on the line easily and we know the intercepts.

So therefore, we can easily work out the slope. And as you can see from the point-slope form, what we need is the slope, which is π‘š, and a point on the line π‘₯ sub one, 𝑦 sub one. It is also worth noting that you might also see the point-slope form as 𝑦 minus 𝐴 equals π‘š multiplied by π‘₯ minus 𝐡, which is exactly the same. They’re just giving the coordinates of the point as 𝐴, 𝐡.

So what we know is that the π‘₯-intercept is negative three and the 𝑦-intercept is four. So therefore, we have two points: negative three, zero and zero, four. So the first thing we want to do is find the slope of our line. And we can do that using a formula. And that formula is π‘š is equal to 𝑦 sub two minus 𝑦 sub one over π‘₯ sub two minus π‘₯ sub one, so the change in 𝑦 over the change in π‘₯.

So to help us use this formula, what we’ve done is labeled our points. So we’ve got π‘₯ sub one, 𝑦 sub one and π‘₯ sub two, 𝑦 sub two. So if we substitute these values in, we get π‘š equals four minus zero over zero minus negative three, which is gonna give us π‘š is equal to four over three or a slope of four-thirds. It’s worth noting that it doesn’t matter which way round we’d labeled our points because it still would’ve given us the same slope.

So great, we now have our slope, and we also know a point on our line. So therefore, we can substitute our information into the point-slope form. We could choose any point along our line. However, I’m just gonna choose here the first point that we’ve got, which is negative three, zero. As we said, any point on our line would work. So when we substitute in our π‘₯ sub one, 𝑦 sub one, what we’re gonna get is 𝑦 minus zero is equal to four-thirds multiplied by π‘₯ minus negative three. So if we take a look at the right-hand side, what we get is 𝑦 equals four-thirds multiplied by π‘₯ plus three. So this is the equation in the point-slope form.

But we could also write it in standard form. So to do this, what we do is multiply the whole thing through by three to give us three 𝑦 equals four multiplied by π‘₯ plus three. Then distributing across parentheses gives us three 𝑦 equals four π‘₯ plus 12. And then, finally, we can subtract four π‘₯, which gives us three 𝑦 minus four π‘₯ equals 12. So therefore, we’ve got the equation in both the point-slope form and standard form.

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