Question

Find the equation for a linear function y where y(6) = 83 and y(9) = 122

Answer 1

`y = 13\, x + 5`.

Explanation:

Start by finding the slope of this line.

In general, if a non-vertical line contains two points,
`(x_(0),\, y_(0))` and
`(x_(1),\, y_(1))`, where
`x_(0) \\e x_(1)`, the slope
`m` of this line would be:

`\displaystyle (y_(1) - y_(0))/(x_(1) - x_(0))`.

In this question, the two points on this line are
`(6,\, 83)` and
`(9,\, 122)`. Therefore, the slope of this line would be:

`\begin{aligned}& (122 - 83)/(9 - 6) \\ =\; & (39)/(3) \\ =\; & 13\end{aligned}`.

Next, find the equation of this line in the point-slope form.

In general, if a non-vertical line of slope
`m` contains a point
`(x_(0),\, y_(0))`, the point-slope form of this line would be:

`\displaystyle y - y_(0) = m\, (x - x_(0))`.

In this question, it was already found that the slope
`m` takes the value
`13`. Two points on this line are given. Using either of them as
`(x_(0),\, y_(0))` would work. For example, with
`(6,\, 83)` as the point, the point-slope equation of this line would be:

`y - 83 = 13\, (x - 6)`.

Finally, rewrite the point-slope form equation of this line into the slope-intercept form.

The slope-intercept equation of a line is of the form
`y = m\, x + b` for slope
`m` and
`y`-intercept
`b`. Rearrange the point-slope equation
`y - 83 = 13\, (x - 6)` into this new form:

`y = 13\, (x - 6) + 83`.

`y = 13\, x - 78 + 83`.

`y = 13\, x + 5`.