Chapter 4, Problem 5CR

To determine

Write a linear function.

Answer to Problem 5CR

The linear function is $f=(-5/3)x+18.$

Explanation of Solution

Given information:

Given values are

  $f(6)=8,f(9)=3.$

So given points are (6,8) and (9,3).

Calculation:

Formula :

In general slope intercept equation of a line or linear function is,

  $y=mx+b$ ….1

where $m$ is the slope of the line and $b$ is the y-intercept.

Finding $m$:

The slope of the line or a linear function is the ration of the change of $y$ over the change in $x$ between any two points on the line .To find the slope between the points (6,8) and (9,3)

  $\begin{array}{c}\text{slope=change}\text{in}y/\text{change}\text{in}x.\\ m=(y_{2-}{y}_1)/(x_2-x_1).\\ m=(3-8)/(9-6).\\ m=-5/3.\end{array}$

Substituting $m=-5/3$ in equation (1),

  $\begin{array}{l}y=(-5/3)\times x+b.\end{array}$ ………(2)

Finding $b$:

Since given value is (6,8) so substituting 6 for $x$ and 8 for $y$ in equation (2).

  $\begin{array}{l}8=(-5/3)\times 6+b.\\ 8=-10+b.\\ b=8+10.\\ b=18.\end{array}$

Substituting the $b=18$ in equation (2),

  $y=(-5/3)x+18$ so,

The linear function with given values (6,8) and (9,3) is $f=(-5/3)x+18.$