Slope Intercept

Once again, we have two racers, one faster than the other.

On the graph, the blue line represents the faster racer and the brown line represents the slower racer.

But notice something important. The brown line does not start at the origin. That means it has a y intercept, a head start.

The line shown in blue has a slope of point one five. The slower racer's line, in brown, has a slope of point zero nine, but once again, it begins higher up on the y axis.

They'll be racing head to head along a five meter track. Because of that head start, this should be a close race.

Graphs are a great way to make predictions, but if we want an even more precise method, we can create equations for each line.

We've learned that we can use y equals mx to describe motion, where y represents the distance in meters, m is the slope or speed, and x is time in seconds.

Let's start with the fast racer. The equation is y equals point one five x.

Now let's predict how long it will take the fast racer to reach five meters. We can plug five in for y.

Five equals point one five x.

We're solving for x, the time. We can ask, five equals point one five times what number?

We can use division to solve for x. Dividing five by point one five gives us thirty three point three three.

That's the number of seconds it will take the faster racer to reach the five meter mark.

Now let's look at the slower racer. Its slope is point zero nine. Y equals point zero nine x.

But that won't work here because its line does not begin at zero. Think about it this way. This equation would mean that when x, the number of seconds, is zero, then y, the distance, would also equal zero. And that just isn't the case here.

We need an equation that accounts for the y intercept, the starting position.

That equation is y equals mx plus b, where b represents the y intercept.

Y equals mx plus b. This equation has a name. It's called slope intercept form.

So for the slow racer, the equation is y equals point zero nine x plus two.

Now we can use this equation to find out how long it will take this racer to reach the five meter finish line. Plug in five for y.

Five equals point zero nine x plus two.

We can think of it this way. Five equals point zero nine times what number plus two?

Okay. So that's a little complicated. Let's make it simpler.

Because the racer is actually starting at a distance of two, what we are really trying to figure out is how long it will take to travel three meters.

If we subtract or cancel out that two from both sides, we're really solving three equals point zero nine x.

Now we can divide three by point zero nine. That gives us thirty three point three three.

And that means that x equals thirty three point three three. That's the number of seconds it will take to reach the finish line.

Now if that number looks familiar, that's because it was exactly what we had calculated for the fast racer. Wow, a perfect tie.

And we can see that when both lines are on the same graph. They intersect or cross paths right at the five meter mark.

Using graphs to make predictions and visualize movement is a great strategy, but using equations gives us another powerful tool. Equations allow us to calculate exact values, like how long it takes to reach five meters, even if the graph isn't perfectly drawn or the scale is hard to read.