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

Kathryn Kingham

A line 𝐿 passes through the points (3, 3) and (−1, 0). Work out the equation of the line, giving your answer in the form 𝑎𝑦 + 𝑏𝑥 + 𝑐 = 0.

03:53

Video Transcript

A line 𝐿 passes through points three, three and negative one, zero. Work out the equation of the line, giving your answer in the form 𝑎𝑦 plus 𝑏𝑥 plus 𝑐 equals zero.

What we’re going to do first, is work out the equation of the line. And after that, we’ll focus on moving it into the form 𝑎𝑦 plus 𝑏𝑥 plus 𝑐.

To solve this problem, we’ll need a few things. Let’s start with finding the slope and the 𝑦-intercept. The formula for finding slope is, the changes in 𝑦 over the changes in 𝑥. And we write that, 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. We plug in these two points for our formula for finding slope. 𝑦 two is zero, minus 𝑦 one, which is three. 𝑥 two is negative one, minus 𝑥 one, which is three. We end up with negative three over negative four. We can simplify our slope to 𝑚 equals three-fourths.

Now that we know our 𝑚 is three-fourths, we can use this slope intercept form to find the intercept of this equation, to find our 𝑏. So we take our point negative one, zero and we plug those values in for 𝑥 and 𝑦. This gives us zero equals three-fourths times negative one plus 𝑏. When I multiply negative one by three-fourths, I get negative three-fourths. To get 𝑏 by itself, I add three-fourths to the right side of the equation. And if I add three-fourths to the right side of the equation, I need to add three-fourths to the left side of the equation. This means that our 𝑦-intercept equals three-fourths, 𝑏 equals three-fourths.

So we use this formula, and we plug in the 𝑚 and 𝑏 that we found. This gives us 𝑦 equals three-fourths 𝑥 plus three-fourths. Now we need to convert the slope intercept form into 𝑎𝑦 plus 𝑏𝑥 plus 𝑐 equals zero. This means that we’ll move everything to the left side of the equation, leaving only zero on the right side of the equation. We start that by subtracting three-fourths 𝑥 from both sides of the equation. This leaves us with 𝑦 minus three-fourths 𝑥 equals three-fourths. Then I can subtract three-fourths from both sides of the equation, which leaves me with 𝑦 minus three-fourths 𝑥 minus three-fourths equals zero.

But this is not our final answer. Because when we work with a form like this, we want 𝑎, 𝑏, and 𝑐 to be integers, which means we don’t wanna have fractions like three-fourths as 𝑏 or 𝑐. To fix this problem, we can multiply our entire form by four. We distribute the four to each term in the equation. Four times 𝑦 equals four 𝑦. Four times negative three-fourths equals negative three. Four times negative three-fourths equals negative three. Four times zero equals zero.

This is the equation of line 𝐿 written as 𝑎𝑦 plus 𝑏𝑥 plus 𝑐. We would write four 𝑦 minus three 𝑥 minus three equals zero.