Question

What is the value of `y` so that the line through `(2,3)` and `(5, y)` has a slope of `-2`?

Answer 1

`y=-3`

Explanation:

Use point-slope form to get a line of equation
`y-3=-2(x-2)`
Put `(5,y)` to the equation
Get `y=-3`

Answer 2

`y_2=-3`

`(y_2-y_1)/(x_2-x_1)=(y_2-3)/(5-2)->( -3-3)/(5-2)`

Explanation:

The slope (gradient) is the amount of up/down for the amount of along as you read from left to right.

Example:

Suppose we hade a slope of 2. This means that for 1 along we go up 2

Suppose we had a slope of -2. This means that for 1 along we go down 2.

Slope is

`color(brown)(("change in y")/("change in x"))color(green)( =(y_("end point")-y_("start point"))/(x_("end point")-x_("start point")))color(blue)(= (y_2-y_1)/(x_2-x_1))`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Solving the question")`

Given:

`"start point"->P_1 ->(x_1,y_1) = (2,3)`
`"end point" color(white)(.)->P_2 ->(x_2,y_2) = (5,y_2)`

`=>(y_2-y_1)/(x_2-x_1) =(y_2-3)/(5-2)=(y_2-3)/3=-2`

Multiply both sides by 3

`=>(y_2-3)xx3/3 = 3xx(-2)`

But `3/3=1`

`=>y_3-3=-6`

Add 3 to both sides

`=>y_2-3+3=-6+3`

`=>y_2+0=-3`

`y_2=-3`