Question Video: Finding the Equation of a Line Passing through Two Points Mathematics

The equation of the straight line passing through the origin and the point (3, −3) is _. [A] 𝑦 = 3 [B] 𝑦 = −4 [C] 𝑥 = 𝑦 [D] 𝑥 + 𝑦 = 0

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Video Transcript

The equation of the straight line passing through the origin and the point three, negative three is blank. Option (A) 𝑦 is equal to three, option (B) 𝑦 is equal to negative four, option (C) 𝑥 is equal to 𝑦, or option (D) 𝑥 plus 𝑦 is equal to zero.

In this question, we’re given some information about a straight line, and we’re asked to find the equation of this straight line. The information we’re given is that our straight line passes through the origin. And we’re also told that it passes through the point three, negative three. And there’s a lot of different methods we could use to answer to this question. However, because we’re specifically asked for the equation of a straight line, let’s do this by using the slope–intercept method of a straight line.

We recall we can represent any nonvertical straight line in the form 𝑦 is equal to 𝑚𝑥 plus 𝑏, where 𝑚 is the slope of our straight line and 𝑏 is the 𝑦-intercept of our straight line. And this is called the slope–intercept form of our straight line. And we don’t need to worry too much about the case where we have a vertical line because we’ll find this out when we calculate the slope of our straight line. If we had a vertical line, we would get an undefined slope. We would have to divide through by zero. So let’s try and write the equation of the straight line given to us in the question in this form.

We need to find two things; we need to find the slope of our straight line and we need to find the 𝑦-intercept of our straight line. Well, first, we can see that we know our straight line passes through the origin. And we know the coordinates of the origin is the point zero, zero. And although it’s not necessary to answer this question, it might be worth sketching this. We know our straight line passes through the origin. And if our straight line passes through the origin, this must also be the 𝑦-intercept of our straight line. So our curve intercepts the 𝑦-axis at the value of zero. In other words, our value of 𝑏 is equal to zero.

We also know that our curve passes through the point three, negative three. We can also plot this point on our graph. And this is enough to sketch our straight line. However, we still need to calculate the slope of this straight line. And to do this, we need to recall what we mean by the slope. The slope is the rise divided by the run. It’s the change in 𝑦 divided by the change in 𝑥. And we have a formula for calculating this for our straight line. We know if our straight line passes through two points 𝑥 sub one, 𝑦 sub one and 𝑥 sub two, 𝑦 sub two, then our value of 𝑚 is 𝑦 sub two minus 𝑦 sub one all divided by 𝑥 sub two minus 𝑥 sub one.

And we’ve already found the coordinates of two points our straight line passes through. It passes through the origin, the point zero, zero, and the point three, negative three. So we can just substitute these values in to find the slope of the straight line given to us in the question. We get 𝑚 is equal to negative three minus zero all divided by three minus zero. This simplifies to give us negative three over three, which we can calculate is negative one. And it’s worth pointing out here this agrees with the sketch we’ve drawn. We can see the slope of this line is in fact negative.

Now, all we need to do is substitute the value of 𝑚 is equal to negative one into the equation for our straight line since we already showed the value of 𝑏 was zero. This gives us the equation 𝑦 is equal to negative 𝑥. However, we can also write this in what’s called the general form of a straight line, that’s in the form 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 is equal to zero, by adding 𝑥 to both sides of our equation. This gives us the equation 𝑥 plus 𝑦 is equal to zero, which was option (D). Therefore, we were able to find the equation of the straight line passing through the origin and the point three, negative three. In the general form of a straight line, this is given by 𝑥 plus 𝑦 is equal to zero.

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