Question

How do you write an equation in slope-intercept form of the line that passes through the points (-2, 6.9) and (-4, 4.6)?

Answer 1

`color(blue)(y = 1.15 x + 9.2 " is the slope - intercept form"`

`color(green)(Slope = m = 1.15, "y-intercept " = 9.2`

Explanation:

Equation of a line, knowing two points on it is given by

`(y-y_1)/(y_2-y_1) = (x-x_1) / (x_2-x_1)`

`(x_1,y_1) = -2, 6.9), (x_2,y_2) = (-4, 4.6)`

`(y - 6.9) / (4.6 - 6.9) = (x +2) / (-4+2)`

`(y - 6.9) / -2.3 = (x + 2) / -2`

`(y - 6.9) = (2.3 * (x + 2))/2, " cross multiplying"`

`y = 1.15 * (x + 2) + 6.9`

`color(blue)(y = 1.15 x + 9.2 " is the slope - intercept form"`

`color(green)(Slope = m = 1.15, "y-intercept " = 9.2`

Answer 2

`y=1.15x+9.2`

Explanation:

Since this is a linear variation of the form `y=mx+b`, any change in `x` will create a proportional change in `y`.
`(-2)-(-4) = 2` and
`(6.9)-(4.6) = 2.3` so
for every 2 changes in `x`, `y` changes by 2.3.
Divide each side by 2, and 1 change in `x` corresponds to 1.15 in `y`, therefore the slope (`m`) must be 1.15.
Now we have the equation `y=1.15x+b`
Before, we said our change in `x` by 2 resulted in a change in `y` of 2.3. Therefore, if we move over right 2 from `(-2, 6.9)`, we reach the point `(0, 9.2)`. Since the `x`-value is 0, this is the y-intercept (`b`)
The equation is `y=1.15x+9.2`