Determine the equation of the line passing through 𝐴: zero, 16 and 𝐵: one, negative nine in slope-intercept form.
So in this question, we’re given the coordinates of two points that lie on a straight line, and we’re asked to determine its equation. We’ve also been asked for the equation in a specific form, slope-intercept form. This is 𝑦 equals 𝑚𝑥 plus 𝑐, where 𝑚 represents the slope of the line and 𝑐 represents the 𝑦-intercept. In order to answer this question, we need to determine both of these values, the slope and the 𝑦-intercept.
So let’s look carefully at the information in the question and the coordinates of the two points we’ve been given that lie on the line. The first of these two points, 𝐴, has coordinates zero, 16. As the 𝑥-coordinate of this point is zero, this is a point on the 𝑦-axis. And therefore, these are the coordinates of the 𝑦-intercept. The 𝑦-coordinate is 16 and this tells us the value of 𝑐 in our equation is also 16. So we can substitute this value of 𝑐 into the equation of our line. So we have that 𝑦 equals 𝑚𝑥 plus 16 for a value of 𝑚 that we now need to calculate.
Remember, 𝑚 represents the slope of the line. If we know the coordinates of two points on the line — which we’ll refer to as 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two — then the slope can be calculated using this formula: 𝑚 is equal to 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. It doesn’t matter which of the two points we think of as 𝑥 one, 𝑦 one and which we think of as 𝑥 two, 𝑦 two. I’m going to choose to subtract the coordinates of point 𝐴 from those of point 𝐵.
So let’s calculate the slope of this line. 𝑚 is equal to 𝑦 two minus 𝑦 one, so that is negative nine minus 16. And then I need to divide by 𝑥 two minus 𝑥 one, so that’s one minus zero. My calculation for the slope simplifies to negative 25 over one, and of course that’s just equal to negative 25. The final step in answering this question is I need to substitute this calculated value of 𝑚 into the equation of the line.
So in doing so, I have that 𝑦 is equal to negative 25𝑥 plus 16. And that’s our answer to this problem, the equation of the line in slope-intercept form.