Question Video: Finding the Equation of a Line through Two Points Giving Your Answer in a Specified Form Mathematics

Video Transcript

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 is equal to zero.

We’ve been given the coordinates of two points that lie on the straight line. The form in which we’re asked to give the equation, 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 equals zero, is the general form of the equation of a straight line. This is called the general equation of a straight line because the equation of every straight line can be written in this form. The values of 𝑎, 𝑏, or 𝑐 may be zero. But it’s still possible to write every straight line in this form whether it’s a diagonal line, a horizontal line, or a vertical line.

To begin this problem though, we’re going to use the slope–intercept form of the equation of a straight line: 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 gives the slope of the line and the value of 𝑏 is its 𝑦-intercept. We should be aware that the value of 𝑏 here isn’t the same as the value of 𝑏 in the general form. These are the letters commonly used though, so we’ll stick with them for now. But we need to be aware that they aren’t the same.

Now, we know that this line passes through the point 𝐵, which has coordinates zero, five, In other words, when 𝑥 is equal to zero, 𝑦 is equal to five. As this point has an 𝑥-coordinate of zero, it lies on the 𝑦-axis. And so we know that the 𝑦-intercept of the line is five.

We therefore determine the value of one of the two unknowns 𝑚 and 𝑏. And we know that the equation of this line in slope–intercept form at the moment is of the form 𝑦 equals 𝑚𝑥 plus five. To find the value of 𝑚, we need to calculate the slope of the line. We recall that the slope of a line is its change in 𝑦 over its change in 𝑥. And if we have the coordinates of two points 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two that lie on the line, the slope can be calculated using the formula 𝑚 equals 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one.

If we allow 𝐴 to be the point 𝑥 one, 𝑦 one and 𝐵 to be the point 𝑥 two, 𝑦 two, we can substitute these values into the slope formula. We have five minus two in the numerator and zero minus negative 10 in the denominator. We need to be especially careful here because a common mistake would be to simply write zero minus 10. Evaluating gives three in the numerator and positive 10 in the denominator. So the slope of the line is three-tenths. We can therefore substitute this value of 𝑚 into the slope–intercept form of the equation of the line. And we have 𝑦 equals three-tenths of 𝑥 plus five.

Now, this is the equation of the line passing through the points 𝐴 and 𝐵. But it isn’t in the form in which we’ve been asked to give our answer. In the general form, we can see that the three terms, the 𝑥-term, the 𝑦-term, and the constant term, are all on the same side of the equation. Although it doesn’t specify this, the values of 𝑎, 𝑏, and 𝑐 in this general form are usually integers. So we also need to deal with the fact that we have a fraction of three-tenths.

We’ll begin then by multiplying every term in our equation by 10. And it gives 10𝑦 is equal to three 𝑥 plus 50. We then want to group all of the terms on the same side of the equation, which we can do by subtracting both three 𝑥 and 50 from each side, giving negative three 𝑥 plus 10𝑦 minus 50 is equal to zero. This is now in the general form of the equation of a straight line. We can see that the value of 𝑎, the coefficient of 𝑥, is negative three. The value of 𝑏, which is the coefficient of 𝑦, is positive 10. And the value of 𝑐, the constant term, is negative 50. So we have our answer.

It’s also worth pointing out that we could’ve chosen to group the terms on the other side of the equation. By subtracting 10𝑦 from each side, we’d have the equation three 𝑥 minus 10𝑦 plus 50 is equal to zero, which is the exact negative of the equation we’ve already found. These equations are equivalent as they can be obtained from the other by multiplying through by negative one. Our answer then, the equation of the line that passes through the points 𝐴 and 𝐵 in its general form, is negative three 𝑥 plus 10𝑦 minus 50 equals zero.