The equation of a line

Find the equation of the line

So you can look at the gradient as being the rate of the increase or decrease of a line. The line in the question is going up, so we can think 'for every 1 along, how many up do we go?'. That is what the gradient is telling us. So if we had a gradient of 2, we would see that if we went along by 1, so our x value increased by 1, we would go up by 2, so our y value increases by 2.
If we look at our line, you chose the points (0,6) and (3,9).
Between these points, our x value has increased by 3, from 0 to 3, and our y value has also increased by 3, from 6 to 9.
The equation for finding the gradient is 'change in y' - so this is how much up or down we have gone, divided by the 'change in x' - which is how far along we have gone.
So we would put in gradient (symbol m):
m = (change in y)/(change in x) = (9-6)/(3-0) = 3/3 = 1
So do you see that our change in the y has been positive 3, from 6 to 9. To find that difference, we did the 9-6 on the top of the fraction.
We can now see that our gradient is 1, which means that for every 1 we go along, we will also go 1 up.
To find the y intercept, we can look at the graph. The y intercept is when the line crosses the y axis. We can see that is when the value for x is 0, and the value of y is 6. So the point (0,6).
Therefore our y intercept is going to be 6.
y = mx +c , so we found that m = 1 and c = 6,
so y = 1x + 6 = x + 6

Using the other equation of y-y1 = m(x-x1) - this is good when it is a bit more complicated and we can't see the y intercept clearly
Let's use the point (3,9), so our x1 value is 3 and our y1 value is 9, m is still 1
y-y1 = m(x-x1)
y-9=1(x-3)
y-9=x-3 [add 9 to both sides]
y=x + 6, which is the same as we had before.

Does that make sense? If you have any more questions, just ask

Thank you so much for the explanation! I understand this now and I can finally finish the last piece of homework I was struggling on! Thank youuu!!! I hope you have a wonderful day/night! You're the best! <33