Lesson Video: Equation of a Straight Line: General Form Mathematics • 9th Grade

Video Transcript

In this video, we will learn how to find and write the equation of a straight line in general form.

There are many forms in which the equation of a straight line can be written, and we may already be familiar with some of them. For example, there’s the slope–intercept form, 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 represents the slope of the line and 𝑏 represents its 𝑦-intercept. Another form in which we can give the equation of a straight line is the point–slope form, 𝑦 minus 𝑦 one equals 𝑚 multiplied by 𝑥 minus 𝑥 one, where 𝑚 represents the slope of the line as before and 𝑥 one, 𝑦 one represents the coordinates of any point that lies on the line.

These two forms of the equation of a straight line are useful in different contexts. For example, from the slope–intercept form, we can identify both the slope and 𝑦-intercept of a straight line at a glance. But these forms do also have their limitations mainly because it isn’t possible to write the equation of every straight line in either of these forms, in particular, vertical lines of equations of the form 𝑥 equals 𝑘 for some constant value of 𝑘. This can be rearranged to 𝑥 minus 𝑘 equals zero. But it isn’t possible to rearrange this equation into either of the two forms we’ve already discussed because it has no 𝑦-term, or at least the coefficient of 𝑦 is zero.

We’ll therefore introduce a third form of the equation of a straight line. This is called the general form because the equation of every straight line can be written in this form. The general form of the equation of a straight line then is 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 equals zero, where 𝑎, 𝑏, and 𝑐 are real constants which may be equal to zero. We note that all three terms are on the same side of the equation, and the right-hand side is equal to zero.

It isn’t required that 𝑎, 𝑏, and 𝑐 are integers, but usually when we give our answers, we do ensure that they are. In the case of vertical lines, which we were concerned about earlier, their equations can be rearranged to 𝑥 minus 𝑘 equals zero. And this is in the general form with 𝑎, the coefficient of 𝑥, equal to one, 𝑏, the coefficient of 𝑦, equal to zero, and 𝑐, the constant term, equal to negative 𝑘.

In the same way, the equations of horizontal lines which we can write as 𝑦 equals some constant 𝑘 two can be rearranged to 𝑦 minus 𝑘 two is equal to zero. And this is also in the general form, this time with 𝑎 equal to zero, 𝑏 equal to one, and 𝑐 equal to negative 𝑘 two. The equation of any diagonal line can also be written in this form. And we’ll see examples of this during the video. Now this form, the general equation of a straight line, is very closely related to the standard form of the equation of a straight line. This is 𝐴𝑥 plus 𝐵𝑦 equals 𝐶, where 𝐴, 𝐵, and 𝐶 are integers and 𝐶 is nonnegative. We must ensure that we are clear on the distinction between the two, which is predominantly in the position of the constant term.

One drawback of the general form of the equation of a straight line is that we cannot directly identify the slope of a line given in this form from its equation. In order to find the slope, we need to rearrange the equation into the slope–intercept form and then read off the coefficient of 𝑥. We’ll demonstrate how to do this in our first example.

A straight line has the equation negative 15𝑥 plus three 𝑦 minus 12 equals zero. What is the slope of this line?

The equation of this straight line has been given in the general form. That’s 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 equals zero with 𝑎 equal to negative 15, 𝑏 equal to positive three, and 𝑐 equal to negative 12. To determine the slope of this line, we can rearrange its equation to the slope–intercept form. That’s 𝑦 equals 𝑚𝑥 plus 𝑏, where the value of 𝑚, the coefficient of 𝑥, is the slope of the line. Now the value of 𝑏 in this equation gives the 𝑦-intercept. And it isn’t necessarily the same as the value of 𝑏 in the general form, but these are the letters usually used, so we’ll stick with them.

So we’ll take the equation we’ve been given, and we’ll rearrange it to make 𝑦 the subject. First, we need to isolate the 𝑦-term, which we can do by adding 15𝑥 and 12 to each side of the equation. That gives three 𝑦 equals 15𝑥 plus 12. We then divide the entire equation by three, which gives 𝑦 equals five 𝑥 plus four. This equation is now in the slope–intercept form. And we can identify that the coefficient of 𝑥 is five. So the slope of the straight line with equation negative 15𝑥 plus three 𝑦 minus 12 equals zero is five.

In the example we just considered, we found the slope of a line by rearranging its equation from the general form to the slope–intercept form. We found that the slope of this line was five, and this came from simplifying the fraction 15 over three. Now, that value of 15 in the numerator is the negative of the coefficient of 𝑥 in the general form, and the value of three in the denominator is the coefficient of 𝑦. This illustrates a general result. If we were to rearrange the general form of the equation of a straight line still in its algebraic form, we would obtain 𝑦 equals negative 𝑎 over 𝑏 𝑥 minus 𝑐 over 𝑏. The slope of this line will be given by the coefficient of 𝑥, which is negative 𝑎 over 𝑏, and so we can state a general result.

The slope of the line 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 equals zero, where 𝑏 is nonzero, is negative 𝑎 over 𝑏. This provides a shortcut for finding the slope of a straight line given in the general form. We can also identify that the 𝑦-intercept of this line is negative 𝑐 over 𝑏, and again 𝑏 must be nonzero. Let’s now consider an example in which we will determine both the 𝑥- and 𝑦-intercepts of a straight line from its general form.

What are the 𝑥-intercept and 𝑦-intercept of the line three 𝑥 plus two 𝑦 minus 12 equals zero?

We’ll recall first that the 𝑥- and 𝑦-intercept of a line are the values of 𝑥 and 𝑦 at which the line intersects that axis. The 𝑥-intercept has a 𝑦-coordinate of zero, and the 𝑦-intercept has an 𝑥-coordinate of zero. To obtain the value of the 𝑥-intercept first then, we can substitute 𝑦 equals zero into the equation of the straight line. This gives three 𝑥 plus two multiplied by zero minus 12 is equal to zero, or simply three 𝑥 minus 12 equals zero. We can then solve this equation to find the value of 𝑥. First, we add 12 to each side, giving three 𝑥 equals 12. And then we divide both sides of the equation by three, giving 𝑥 equals four. And so the 𝑥-intercept of this straight line is four.

To find the 𝑦-intercept, we substitute 𝑥 equals zero into the equation of the line. And this gives three multiplied by zero plus two 𝑦 minus 12 equals zero. That’s just two 𝑦 minus 12 equals zero. And then we solve for 𝑦. First, we add 12 to each side of the equation and then we divide by two, giving 𝑦 is equal to six. And so the 𝑦-intercept of this line is six. We can state then that the 𝑥-intercept and 𝑦-intercept of the line three 𝑥 plus two 𝑦 minus 12 equals zero are four and six, respectively.

We’ve now seen how to find the slope and the 𝑥- and 𝑦-intercepts of a straight line from its equation given in the general form. Let’s now consider how we can find the equation of a straight line in the general form given its slope and its 𝑦-intercept.

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

We’re given both the slope and the 𝑦-intercept of this straight line. We can therefore write down its equation by recalling first the slope–intercept form of the equation of a straight line. That’s 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 represents the slope of the line and 𝑏 represents its 𝑦-intercept. The slope of this straight line is three over two, and the 𝑦-coordinate of its 𝑦-intercept is three. So that’s the value of 𝑏. Substituting three over two for 𝑚 and three for 𝑏 then, we have 𝑦 equals three over two 𝑥 plus three.

Now, this is the equation of the line we’re looking for, but we’ve been asked to give our answer in a specific form: 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 equals zero. This is the general form of the equation of a straight line. We therefore need to rearrange the equation we’ve written. We note first that in the general term, all three terms are on the same side of the equation. So we’ll begin by subtracting the 𝑦-term in our equation from both sides. When we do this, we obtain zero is equal to three over two 𝑥 minus 𝑦 plus three. And we can, of course, write the two sides of the equation the other way around.

Now, although this isn’t strictly necessary, it’s usual that the values of 𝑎, 𝑏, and 𝑐 in the general form are integers. We have a coefficient of 𝑥 which is a fraction. So in order to eliminate the denominator of two, we can multiply every term in our equation by two. This gives three 𝑥 minus two 𝑦 plus six is equal to zero. This is now in the general form with 𝑎, the coefficient of 𝑥, equal to three, 𝑏, the coefficient of 𝑦, equal to negative two, and 𝑐, the constant term, equal to positive six. So we found that the equation of the line with slope three over two and 𝑦-intercept zero, three in the general form is three 𝑥 minus two 𝑦 plus six equals zero.

Let’s now consider an example in which we find the equation of a straight line in general form, given a different set of information. This time, the information given will be the coordinates of two points that lie on the line.

Find the equation of the line that passes through the points 𝐴 negative 10, two and 𝐵 zero, five, giving your answer in the form 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 equals zero.

We’ve been given the coordinates of two points that lie on this line. The form we’ve been asked to give our answer in is the general form of the equation of a straight line: 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 equals zero, where 𝑎, 𝑏, and 𝑐 are real constants. To begin with, though, we’ll approach this problem using the slope–intercept form of the equation of a straight line. That’s 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 is the slope of the line and 𝑏 is its 𝑦-intercept. We should be aware that this value of 𝑏 isn’t necessarily the same as the value of 𝑏 in the general form, but these are the letters that are generally used for both.

Now we know that this line passes through the point 𝐵, which has coordinates zero, five. This is the point on the 𝑦-axis. And so the 𝑦-intercept of the line is five. This means that the value of 𝑏 in our slope–intercept form is also five. And so the equation is now of the form 𝑦 equals 𝑚𝑥 plus five. And we need to calculate the slope of the line.

To do this, we recall that the slope of a line is its change in 𝑦 divided by its change in 𝑥. If we have two points which lie on a line with coordinates 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two, then the slope can be calculated as 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. We can let 𝐴 be the point 𝑥 one, 𝑦 one and 𝐵 be the point 𝑥 two, 𝑦 two and then substitute the values for these coordinates into the slope formula. That gives five minus two in the numerator and zero minus negative 10 in the denominator. And we need to be especially careful here to ensure we include both of those negative signs.

That simplifies to three over 10 because zero minus negative 10 in the denominator is zero plus 10, which is 10. Substituting 𝑚 equals three-tenths into the slope–intercept form then, we have 𝑦 equals three-tenths of 𝑥 plus five. We need to rearrange this equation, though, so that it is in the general form. And to do this, we need to collect all of the terms on the same side of the equation. Subtracting three-tenths of 𝑥 and five from each side, we have negative three-tenths of 𝑥 plus 𝑦 minus five equals zero.

Now it’s also usual that in the general form, the values of 𝑎, 𝑏, and 𝑐 are integers, so we’ll multiply the entire equation by 10, which gives negative three 𝑥 plus 10𝑦 minus 50 equals zero. And this is in the general form with 𝑎 equal to negative three, 𝑏 equal to 10, and 𝑐 equal to negative 50. Now we could have grouped the terms on the other side of the equation if after this stage here we had subtracted 𝑦 from each side and then multiplied by 10. This would’ve given three 𝑥 minus 10𝑦 plus 50 equals zero, which is the exact negative of our equation, and therefore an alternative answer also in its general form. Our answer is negative three 𝑥 plus 10𝑦 minus 50 equals zero.

In our final example, we’ll find the equation of a straight line in the general form, this time given its 𝑥- and 𝑦-intercept.

Determine the equation of the line which cuts the 𝑥-axis at four and the 𝑦-axis at seven.

Although it isn’t entirely necessary, we may find it helpful to sketch this straight line. It cuts the 𝑥-axis at four, and it cuts the 𝑦-axis at seven. So it looks something like this. We can see that this line slopes downwards from left to right, and so it has a negative slope. This will give us one way of checking whether our answer, when we reach it, seems reasonable. To answer this question, we’ll begin by using the slope–intercept form of the equation of a straight line: 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 is the slope of the line and 𝑏 is its 𝑦-intercept. We already know the value of 𝑏 because it was given in the question. We were told the line cuts the 𝑦-axis at seven. So our line is 𝑦 equals 𝑚𝑥 plus seven. And we need to calculate its slope.

The slope of a straight line passing through the two points with coordinates 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two is 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. Using the coordinates of the 𝑥- and 𝑦-intercepts then, which are four, zero and zero, seven, we can calculate the slope of this line. It’s zero minus seven over four minus zero, which simplifies to negative seven over four. This is a negative value, so we can be reassured that what we’ve done so far is correct. The equation of this line, then, is 𝑦 equals negative seven over four 𝑥 plus seven.

Now we could leave our answer in this form, but it looks a little unfriendly because we have a quotient. We can find an equivalent form of this equation by rearranging to the general form. We’ll begin by multiplying every term in the equation by four, giving four 𝑦 equals negative seven 𝑥 plus 28. Next, we’ll group all of the terms on the same side of the equation, choosing the left-hand side so that the coefficients of both 𝑥 and 𝑦 are positive. So adding seven 𝑥 and subtracting 28 from each side, we have seven 𝑥 plus four 𝑦 minus 28 is equal to zero. This is the equation of this straight line in its most general form. That’s 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 equals zero, where 𝑎, 𝑏, and 𝑐 are real constants.

Let’s now summarize the key points from this video. The general form of the equation of a straight line is 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 equals zero, where 𝑎, 𝑏, and 𝑐 are real constants, which are usually integers. This is called the general form because every straight line, whether it’s diagonal, vertical, or horizontal, can be represented by an equation in this form. Provided 𝑏 is nonzero, the slope of a line given in general form is negative 𝑎 over 𝑏, and the 𝑦-intercept is negative 𝑐 over 𝑏. We also saw that we can find the equation of a straight line in the general form given different sets of information about the line and that we can rearrange the equation of a straight line given in any other form to the general form using algebraic manipulation.