A line 𝐿 passes through the points two, three and negative two, five. Work out the equation of the line, giving your answer in the form 𝑦 equals 𝑚𝑥 plus
The form that the question asked as to write the equation in for the line 𝐿 is
actually this one here, which is the slope-intercept form. And we’re gonna have a little look at why it’s called that and use it to help us
solve the problem.
When using 𝑦 equals 𝑚𝑥 plus 𝑐, we can see that 𝑚 is equal to the slope. So our coefficient of 𝑥 is the slope of the line and the 𝑐 is the 𝑦-intercept. So therefore, it’s the slope-intercept form.
We’re gonna start by looking at the slope or 𝑚, where 𝑚 is actually the change in
𝑦 divided by the change in 𝑥. 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, we’ve actually got a little formula. And this is 𝑚 is equal to 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one cause it’s
actually calculating the difference between the coordinates.
Great! So we’re gonna use this to work out the slope on this line 𝐿. So we have our two points two, three and negative two, five. And the first thing we’re gonna do is actually label coordinates, a something I’d
probably do until you’re really kinda confident with doing this. It helps you to see what you need to put into the formula.
So now we’ve actually labelled our coordinates, we’ve got 𝑥 one, 𝑦 one, 𝑥 two, 𝑦
two, we can substitute them into our formula to find our slope. So first of all, we’ve got 𝑦 two, which is five, minus 𝑦 one, which is three and
that’s divided by negative two, which is 𝑥 two, minus two, which is 𝑥 one.
Fantastic, now we’re 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 𝑚 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 𝐿.
The first thing we can do is we’re gonna actually substitute our 𝑚 value into our 𝑦
equals 𝑚𝑥 plus 𝑐. Now, in order for us to complete our equation, we need to know the value of 𝑐. We can find 𝑐 by using one of our pairs of coordinates. To do this, we’re gonna use this first pair of coordinates two, three and we’re gonna
substitute our values for 𝑥 and 𝑦 in from our coordinates two, three.
So first of all, we get 𝑦 is equal to three. So this gives us three equals and then we’re gonna substitute in two for our value of
𝑥. So it’s gonna give us negative a half multiplied by two plus 𝑐. Great! So now let’s simplify this and solve to find 𝑐. First of all, we’ve done negative a half multiplied by two, which gives us negative
one. So we’ve got three equals negative one plus 𝑐. So therefore, we get 𝑐 is equal to four.
Great! So we’ve now found our 𝑚, so our slope, and our 𝑐, our 𝑦-intercept. So we can now put these back into our 𝑦 equals 𝑚𝑥 plus 𝑐 to give us the equation
for 𝐿, which is 𝑦 equals negative a half 𝑥 plus four.
Okay, we found the equation of 𝐿, but we’re always have to check to make sure that
all the calculations are correct. And to do that, we’re actually gonna use the coordinates of our second point. And the way we’re gonna do that is by substituting in our 𝑥-value, which is negative
two, and we’re gonna see if it gives out the correct 𝑦-value.
So here, I’ve done that, so 𝑦 equals negative a half multiplied by negative two, so
that’s the 𝑥-value we’ve substituted in plus four, which is gonna give us 𝑦 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 𝑦 is
equal to five. Great! So now we check that with the 𝑦-coordinate. Yes, they’re both five. So now, we know we’ve checked our answer and our equation of 𝐿 is correct.