Write an equation representing the line shown.
The equation of a line is 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 is your slope or the steepness of the line, and 𝑏 is the 𝑦-intercept where we cross the 𝑦-axis. And then 𝑦 and 𝑥 just remain there in the equation. So we need to find 𝑚 and 𝑏, and then this will complete our equation of the line shown.
So first let’s begin with 𝑏. Where do we cross the 𝑦-axis? We cross it here at five. So 𝑥 is zero and then our line is at five. So we care about the 𝑦-value, five. So we can replace 𝑏 with five.
Now next is our slope, the steepness of the graph. Slope is equal to how much we rise divided by how much we run. Now what does that really mean? That means the change vertically divided by the change horizontally. So if we just start where we cross the 𝑦-axis at five, until we get to another specific point that we actually went through on our graph, we can count how much we changed vertically divided by how much we changed horizontally. Now going up will be positive for the vertical and going down would be negative for the vertical. And with horizontal, going to the right would be positive and going left would be negative.
So here are a few points that we go through. So how did we get from each point to the following point? Well, we had to go up one and then we had to go right two. So let’s check for our next point as well, up one right two. And this will stay constant through the whole graph. Now we chose to go to the right. Let’s say you went to the left of where it crossed the 𝑦-axis. We would still get one-half, and here’s why. We would go down one which is negative one, and then we would go left two which is negative two. And two negatives will cancel. So like we said before, we still get the one-half.
So let’s plug in one-half for 𝑚. 𝑦 equals one-half 𝑥 plus five would be our equation. It can also be written with 𝑥 being multiplied to the one-half, so 𝑥 divided by two plus five. Either equation would work.
So again, our final answer would be 𝑦 equals 𝑥 divided by two plus five.