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A line goes through the points (-1, 6)

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and (5, 4). What is the equation of the line?

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Let's just try to visualize this. So that is my

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x axis. And you don't have to draw it

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to do this problem but it always help to visualize

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That is my y axis. And the first point is (-1,6)

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So (-1, 6). So negative 1 coma, 1, 2, 3, 4 ,5 6.

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So it's this point, rigth over there, it's (-1, 6).

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And the other point is (5, -4). So 1, 2, 3, 4, 5.

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And we go down 4, So 1, 2, 3, 4

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So it's rigth over there. So the line connects them

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will looks something like this. Line will

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draw a rough approximation.

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I can draw a straighter than that. I will draw a doted line maybe

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Easier do doted line.

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So the line will looks something like that.

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So let's find its equation. So good place to start

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is we can find its slope.

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Remember, we want, we can find the equation

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y is equal to mx plus b.

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This is the slope-intercept form where m is the slope and b is the y-intercept.

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We can first try to solve for m. We can find the slope of this line.

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So m, or the slope is the change in y over the change in x.

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Or, we can view it as the y value of our end point

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minus the y value of our starting point over

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the x-value of our end point minus the x-value of our starting point.

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Let me make that clear. So this is equal to

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change in y over change in x wich is the same thing as

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rise over run wich is the same thing as the y-value of

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your ending point minus the y-value of your starting point.

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This is the same exact thing as change in y and that over

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the x value of your ending point minus the x-value of your starting point

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This is the exact same thing as change in x.

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And you just have to pick one of these as the starting point

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and one as the ending point. So let's just make this over here

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our starting point and make that our ending point.

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So what is our change in y?

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So our change in y, to go we started at y

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is equal to six, we started at y is equal to 6.

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And we go down all the way to y is equal to negative 4

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So this is rigth here, that is our change in y

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You can look at the graph and say, oh,

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if I start at 6 and I go to negative 4 I went down 10.

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or if you just want to use this formula here it will give you the same thing

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We finished at negative 4, we finished at negative 4

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and from that we want to subtract, we want to subtract 6.

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This right here is y2, our ending y and this is our beginning y

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This is y1. So y2, negative 4 minus y1, 6.

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or negative 4 minus 6.

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That is equal to negative 10.

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And all it does is tell us the change in y

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you go from this point to that point

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We have to go down, our rise is negative

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we have to go down 10. That's where the negative 10 comes from.

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Now we just have to find our change in x.

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So we can look at this graph over here.

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We started at x is equal to negative 1 and we go all the way

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to x is equal to 5. So we started at x is equal to negative 1,

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and we go all the way to x is equal to 5.

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So it takes us one to go to zero and then five more.

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So are change in x is 6.

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You can look at that visually there or

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you can use this formula same exact idea,

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our ending x-value, our ending x-value is 5

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and our starting x-value is negative 1.

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5 minus negative 1. 5 minus negative 1 is the same thing as

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5 plus 1. So it is 6. So our slope here is negative 10 over 6.

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wich is the exact same thing as negative 5 thirds.

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as negative 5 over 3 I devided the numerator and the denominator by 2.

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So we now know our equation will be

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y is equal to negative 5 thirds, that's our slope,

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x plus b. So we still need to solve for y-intercept to get our equation.

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And to do that, we can use the information that we know

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in fact we have several points of information

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We can use the fact that the line goes through the point (-1,6)

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you could use the other point as well.

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We know that when is equal to negative 1,

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So y is eqaul to 6. So y is equal to six when x is equal to negative 1

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So negative 5 thirds times x, when x is equal to negative 1

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y is equal to 6. So we literally just substitute this x and y value

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back into this and know we can solve for be.

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So let's see, this negative 1 times negative 5 thirds.

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So we have 6 is equal to positive five thirds plus b.

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And now we can subtract 5 thirds from both sides of this equation.

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so we have subtracted the left hand side.

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From the left handside and subtracted from the rigth handside

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And then we get, what's 6 minus 5 thirds.

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So that's going to be, let me do it over here

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We take a common denominator. So 6 is the same thing as

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Let's do it over here. So 6 minus 5 over 3 is the same thing as

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6 is the same thing as 18 over 3 minus 5 over 3

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6 is 18 over 3. And this is just 13 over 3. And this is just 13 over 3.

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And then of course, these cancel out. So we get

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b is equal to 13 thirds. So we are done. We know

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We know the slope and we know the y-intercept.

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The equation of our line is y is equal to

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negative 5 thirds x plus our y-intercept which is 13

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which is 13 over 3. And we can write these as mixte numbers.

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if it's easier to visualize. 13 over 3 is four and 1 thirds.

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So this y-intercept right over here.

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this y-intercept right over here. That's 0 coma 13 over 3 or

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0 coma 4 and 1 thirds. And even with my very roughly drawn diagram

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it those looks like this. And the slope negative 5 thirds

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that's the same thing as negative 1 and 2 thirds.

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You can see here the slope is downward because the slope is negative.

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It's a little bit steeper than a slope of 1.

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It's not quite a negative 2. It's negative 1 and 2 thirds.

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if you write ithis as a mixt number.

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So, hopefully, you found that entertaining.