How to Find a Line Equation from a Graph: Gradient and Intercept Explained

in today's video we're looking at how to find the equation of a line when you're given that line on a graph like any of

these three the key idea is that we can write straight line equations in the form of y

equals mx plus c where m is the gradient of the line so how steep it is

and c is the y-intercept which is the point where the line crosses the y-axis

so all you need to do is find the gradient and the y-intercept of a line and plug those values into the equation

to see how this works let's start with this question where we need to find the equation of

the line on this graph the easiest thing to spot is that the line crosses the y-axis at y equals 2

which tells us that c must be 2. the gradient though is a bit trickier and there's a few different ways to find

it but in this video we're going to use the equation change in y

divided by change in x which just means how much the y values change by

divided by how much the x values change by so to use this we pick any two points

along the line for example these two label them point one and point two and then draw vertical and horizontal

arrows between them so that we can form a right angled triangle then to find our change in y we can see

that the vertical arrow goes from positive five all the way down to negative four

which is a change of minus nine so our change in y is minus nine in the same way the horizontal arrow

goes from x equals negative one over to x equals two which gives us a change in x of positive

three this means that to find our gradient we just do the change in y of minus nine

divided by the change in x of three which will give us negative three so m in our equation is minus three

and so to finish we just take our equation y equals mx plus c and plug in the minus 3 as m

and the 2 as c which will give us y equals minus 3x plus 2 as the equation of our line

before we finish let's try the same thing for this one the first thing you want to do with all

of these questions is write out y equals mx plus c because that's the general format that

we're going to try and put our equation into then to find c we need to look for the

y-intercept which is down here at negative three so you know that c is negative three

and next you need to find the gradient which requires us to find the change in y

and divide it by the change in x so for that you need to pick two points along the line

for example 0 negative three and four negative two label them both

and then draw the triangle thing so if we go from point one to point two and start with the y's

we can see that we've gone up by one so our change in y is just one and then if we look at the x values we

can see that we've gone along by four so our change in x is four this means that the gradient will be one

divided by four which can't be simplified so m is just one quarter

then to finish we just plug the c equals negative three and m equals a quarter

into our y equals mx plus c equation to get a final answer of y equals one quarter x minus three

anyway that's everything for this video so hope it all made sense and cheers for watching