Question

What is the equation of the line passing through `(3,7)` and `(-8,12)`?

Answer 1

It is important that you be more specific in your question. I am assuming you mean a 'strait line graph'.

`color(green)(y= -5/11x+92/11)`
You did not request the y and x intercept so not calculated

Explanation:

`color(green)("The standard (std) form of this type of plot is")`

`color(green)(y=mx+c)`

`color(blue)(~~~~~~~~~~~~ "Pre-amble"~~~~~~~~~~~~~~~~~)`

`color(brown)(y " is the dependent variable as it is the outcome of and thus")`
`color(brown)("controlled by what is on the right of the =")`

`color(brown)(x" is the independent variable as it can take on any value you")`
`color(brown)("chose.")`

`color(brown)(m color(white)(x) "is the gradient of the 'curve'. Yes it is mathematically correct")`
`color(brown)("to call a strait line plot a curve. People do not tend to")`
`color(brown)("though!")`

`color(blue)(~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~)`

`color(green)("We start by determining the gradient. We then")`
`color(green)("substitute a pair of the given co-ordinates (Ordered pairs)")`
`color(green)("to find the value of the constant.")`

`color(blue)("Find the gradient")`

`m = ("change in up or down")/("change in along") -> (y_2-y_1)/(x_2-x_1)`

Let `(x_1,y_1) = ( 3,7)`
the left most pair chosen to be so as you listed them first!

Let `(x_2,y_2)=(-8,12)`

Then `m = (y_2-y_1)/(x_2-x_1) = (12-7)/((-8)-3) = 5/(-11)`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Find the constant")`

Using ; `y=mx+c`

then for`color(white)(xx) (x_1,y_1) -> y_1=mx_1+c`

`=> 7=(-5/11)3+c`

`c= 7+15/11 = 8 4/11 " or " 92/11`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Putting it all together!")`

Thus, we now have what we need: the gradient and the constant

The equation is : `y= -5/11x+92/11`

`color(green)("Fractions are precise:::: Decimals are less so!!! Use fractions in")`
`color(green)("preference unless specifically instructed otherwise!!!!!!!")`