### Video Transcript

A line ๐ฟ passes through the points two, three and negative two, five. Work out the equation of the line, giving your answer in the form ๐ฆ equals ๐๐ฅ plus
๐.

The form that the question asked as to write the equation in for the line ๐ฟ is
actually this one here, which is the slope-intercept form. And weโre gonna have a little look at why itโs called that and use it to help us
solve the problem.

When using ๐ฆ equals ๐๐ฅ plus ๐, we can see that ๐ is equal to the slope. So our coefficient of ๐ฅ is the slope of the line and the ๐ is the ๐ฆ-intercept. So therefore, itโs the slope-intercept form.

Weโre gonna start by looking at the slope or ๐, where ๐ is actually the change in
๐ฆ divided by the change in ๐ฅ. So what that means is how much the graph goes up or down divided by how much the
graph goes left or right. Okay, to help us do that, weโve actually got a little formula. And this is ๐ is equal to ๐ฆ two minus ๐ฆ one over ๐ฅ two minus ๐ฅ one cause itโs
actually calculating the difference between the coordinates.

Great! So weโre gonna use this to work out the slope on this line ๐ฟ. So we have our two points two, three and negative two, five. And the first thing weโre gonna do is actually label coordinates, a something Iโd
probably do until youโre really kinda confident with doing this. It helps you to see what you need to put into the formula.

So now weโve actually labelled our coordinates, weโve got ๐ฅ one, ๐ฆ one, ๐ฅ two, ๐ฆ
two, we can substitute them into our formula to find our slope. So first of all, weโve got ๐ฆ two, which is five, minus ๐ฆ one, which is three and
thatโs divided by negative two, which is ๐ฅ two, minus two, which is ๐ฅ one.

Fantastic, now weโre gonna simplify this to get our slope, which is two over negative
four, which again we can simplify further by dividing the numerator and the
denominator by two, which gives us ๐ is equal to negative a half. So fantastic, we found our slope and now we can go onto actually finding the equation
of our line ๐ฟ.

The first thing we can do is weโre gonna actually substitute our ๐ value into our ๐ฆ
equals ๐๐ฅ plus ๐. Now, in order for us to complete our equation, we need to know the value of ๐. We can find ๐ by using one of our pairs of coordinates. To do this, weโre gonna use this first pair of coordinates two, three and weโre gonna
substitute our values for ๐ฅ and ๐ฆ in from our coordinates two, three.

So first of all, we get ๐ฆ is equal to three. So this gives us three equals and then weโre gonna substitute in two for our value of
๐ฅ. So itโs gonna give us negative a half multiplied by two plus ๐. Great! So now letโs simplify this and solve to find ๐. First of all, weโve done negative a half multiplied by two, which gives us negative
one. So weโve got three equals negative one plus ๐. So therefore, we get ๐ is equal to four.

Great! So weโve now found our ๐, so our slope, and our ๐, our ๐ฆ-intercept. So we can now put these back into our ๐ฆ equals ๐๐ฅ plus ๐ to give us the equation
for ๐ฟ, which is ๐ฆ equals negative a half ๐ฅ plus four.

Okay, we found the equation of ๐ฟ, but weโre always have to check to make sure that
all the calculations are correct. And to do that, weโre actually gonna use the coordinates of our second point. And the way weโre gonna do that is by substituting in our ๐ฅ-value, which is negative
two, and weโre gonna see if it gives out the correct ๐ฆ-value.

So here, Iโve done that, so ๐ฆ equals negative a half multiplied by negative two, so
thatโs the ๐ฅ-value weโve substituted in plus four, which is gonna give us ๐ฆ is
equal to one plus four because negative a half multiplied by negative two; negative
multiplied by negative gives us positive, which gives us a final answer of ๐ฆ is
equal to five. Great! So now we check that with the ๐ฆ-coordinate. Yes, theyโre both five. So now, we know weโve checked our answer and our equation of ๐ฟ is correct.