Question

How do you write out the equation of the line passing through `(1,5) (6,15)`?

Answer 1

`y=2x+3`

Explanation:

Use point-slope and slope formula.

We first need to find the slope by finding the change in y over the change in x. This would be `(15-5)/(6-1)=10/5`, which gives us `2`.

Now if we look at point slope formula, it is `y-y_1=m(x-x_1)` where `x_1` and `y_1` are the x and y coordinates of a given point. Let's just say we plug in `(1,5)` for it and the `2` for `m` which is slope.

`y-5=2(x-1)`

Add `5` to both sides.

`y\cancel(-5)\cancel(\color(indianred)(+5))=2(x-1)\color(indianred)(+5)`

Distribute `2`

`y=\color(navy)(2x-2)+5`

Add `-2` and `5`

`y=2x+\color(olive)(3)`

Answer 2

`y=2x+3`

Explanation:

Apply slope formula

• use formula `m=(y_2-y_1)/(x_2-x_1)` given two points `(x_1,y_1)` and `(x_2,y_2)` (doesn't matter which one comes first)
  the slope you get should be `\color(olive)(m)=10/5=2/1=\color(olive)(2)`

Apply point-slope formula

• now plug-in one of the points for the formula `y-y_1=m(x-x_1)`,
  which will now become `y-y_1=\color(olive)(2)(x-x_1)`.
  here, `\color(indianred){(x_1,y_1)}` can be either of the points.
  for easier explanation, used `\color(indianred){(1,5)}`

Working it out

step A `\color(maroon)(rightarrow)`plugging in: `y-\color(indianred)(5)=2(x-\color(indianred)(1))`
step B `\color(maroon)(rightarrow)`distributing: `y-5=\color(seagreen)(2x-2)`
step C `\color(maroon)(rightarrow)` `\color(teal)(5)` added to each side: `y\cancel(-5)\cancel(\color(teal){+5})=2x-2\color(teal)(+5)`
step D `\color(maroon)(rightarrow)`simplifying like terms: `y=2x+\color(green)(3)`

• therefore the equation is:
  `\color{cornflowerblue}{y=2x+3}`