Let’s start from a very simple equation:
y = x
From the graph of this equation, you can tell that it has a slope of 1 because it rises 1 for every 1 we run.
But we need a slope of 2. This means that we need a line that rises 2 for every 1 you run. We have to double the steepness of our line.
To double the y for every x that we run, we change our equation to:
y = 2x
You can graph this equation to verify that it rises 2 for every 1 we run.
But this line does not pass through point (3,1). We know that because, when x is 3, y is 6 (not 1).
By subtracting 5, we will make every y coordinate of our line go down 5 units. And we do want that. So we try the equation:
y = 2x - 5
Let’s try this equation with x=2, x=3, and x=4.
Say x=2. Then y = 2(2) - 5 = 4 - 5 = -1
Say x=3. Then y = 2(3) - 5 = 6 - 5 = 1
Say x=4. Then y = 2(4) - 5 = 8 - 5 = 3
From this, we notice two things. The first one is that our line rises 2 for every 1 we run. So it still has a slope of 2. The second one is that when x=3, y=1. This tells us that our line passes through (3,1).
So, our answer is:
y = 2x - 5
