Question Video: Finding the Equation of a Line in General Form Mathematics

Video Transcript

Write the equation of the line with slope three over two and 𝑦-intercept zero, three in the form 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 is equal to zero.

In this question, we’re asked to find the equation of a straight line. We’re asked to give our answer in the form 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 is equal to zero. And this type of equation for a straight line has a special name. It’s called the general form for the equation of a straight line. So to find this, let’s see what information we’re given about our straight line. We’re told the value of the slope of our straight line is three over two. And we’re also given the 𝑦-intercept of our straight line. That’s the point zero, three.

And we know a form of a straight line which uses both the slope and the 𝑦-intercept. It’s called the slope–intercept form of a straight line. That’s the equation 𝑦 is equal to 𝑚𝑥 plus 𝑘, where 𝑚 is the slope of our straight line and 𝑘 is the value of the 𝑦-intercept. And it’s worth pointing out something here. Usually, in the slope–intercept form, the value of 𝑘 is written as 𝑏 or 𝑐. However, in this case, we’ve used both 𝑏 and 𝑐 in the general form of the equation of our line, so we’ll just rename this 𝑘.

And in fact, we already know the slope of our straight line is three over two. We’re told this in the question. And we’re also told the value of our 𝑦-intercept. This is the point zero, three. So the 𝑦-intercept of our point is going to be equal to three. So in the slope–intercept form of our straight line, we’ll set 𝑚 equal to three over two and 𝑘 equal to three. This gives us the equation 𝑦 is equal to three over two 𝑥 plus three. But remember, we’re asked to give our answer in the general form of the equation of a straight line. That’s the form 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 is equal to zero.

So we have a few options. We’re gonna start by multiplying our equation through by two. This is not technically necessary. However, it will simplify our equation. We need to multiply every term in this equation by two. Three over two times two is three, and three times two is six. So we get two 𝑦 is equal to three 𝑥 plus six. Now, we’ll just subtract two 𝑦 from both sides of this equation. And this gives us our final answer of three 𝑥 minus two 𝑦 plus six is equal to zero.

However, there is something worth pointing out. This is not the only answer we could’ve given. When we had two 𝑦 is equal to three 𝑥 plus six, we subtracted two 𝑦 from both sides of the equation. But we could’ve also subtracted three 𝑥 and subtracted six from both sides of our equation. This would then give us the equation negative three 𝑥 plus two 𝑦 minus six is equal to zero. And this is in the form 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 is equal to zero, and it represents the same straight line.

And in fact, we can see that these two equations represent the same line because if we multiply either through by negative one, we get the other equation. In fact, we can multiply either equation through by any nonzero constant, and we’ll get an equation for the same straight line in the general form. But usually, as a rule of thumb, we try to give our values of 𝑎, 𝑏, and 𝑐 as integers and we try to make our value of 𝑎 positive.

But any other solution of this form is also valid. Therefore, we were able to show the equation of the straight line with slope three over two and 𝑦-intercept zero, three in the form 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 is equal to zero is given by three 𝑥 minus two 𝑦 plus six is equal to zero.