Video: Finding the Equation of a Line given Two Points on the Line

Bethani Gasparine

A line 𝐿 passes through the points (1, 1) and (−5, −1). Work out the equation of the line, giving your answer in the form 𝑦 = 𝑚𝑥 + 𝑏.

02:34

Video Transcript

A line 𝐿 passes through the point one, one and negative five, negative one. Work out the equation of the line giving your answer in the form 𝑦 equals 𝑚𝑥 plus 𝑏. 𝑚 is your slope and 𝑏 is the 𝑦-intercept. To find the slope, there’s a formula also known as 𝑚 where we take 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one.

Now in order to find the 𝑦-intercept, if we had a graph, it would just be where we crossed the 𝑦-axis. Now since we don’t, once we find the slope, we can use the point one, one or negative five, negative one and plug it in for 𝑥 and 𝑦 and then we can solve for 𝑏 at the end.

So let’s first begin by finding the slope of this line. So here we have our two points: one, one and negative five, negative one. And now we’ll plug it into our formula. So we have 𝑦 two minus 𝑦 one, so negative one minus one, over 𝑥 two minus 𝑥 one, negative five minus one, which is negative two over negative six. And the two negatives cancel to make a positive and two-sixths reduces to one-third. So our slope is one-third.

So taking the equation of the line and plugging in one-third for 𝑚, now we have a 𝑦 and 𝑥 and a 𝑏 left. Now the 𝑦 and 𝑥, we can actually substitute in one, one or negative five, negative one. Let’s go ahead and use the point one, one. So using one to plug in for 𝑥, one to plug in for 𝑦, and one-third to plug in for 𝑚 — plugging one in for 𝑥 and one in for 𝑦 and one-third in for 𝑚 — that means we have everything except for 𝑏. This allows us to solve for 𝑏. One-third times one is just one-third. So in order to solve for 𝑏, we need to subtract one-third from both sides of the equation. This means on the right-hand side, the one-third would cancel, but on the left we have one minus one-third.

In order to subtract fractions, they need to have a common denominator. So one is the same thing as three over three. So we can replace one with three-thirds. So three-thirds minus one-third, we subtract the numerators and we keep the denominators. So 𝑏 is equal to two-thirds. Therefore, plugging in one-third for 𝑚, our slope, and two-thirds for 𝑏, our 𝑦-intercept, we get the equation 𝑦 equals one-third 𝑥 plus two-thirds.

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