Chapter 1.1, Problem 35E

To determine

To find: The value of $m$ and $b$ .

Answer to Problem 35E

The value of $m$ and $b$ are $\frac{7}{2}$ and $-\frac{3}{2}$ respectively.

Explanation of Solution

Given information: The given table for a linear function is as follows.

$x$	$f\left(x\right)$
$1$	$2$
$3$	$9$
$5$	$16$

Calculation:

Consider two points $\left(1,2\right)$ , and $\left(3,9\right)$ .

The formula to find the slope of a line passing through two points is given by,

  $m=\frac{y_2-y_1}{x_2-x_1}$

Substitute $1$ for $x_1$ , $2$ for $y_1$ , $3$ for $x_2$ , and $9$ for $y_2$ in the above formula.

  $\begin{array}{c}m=\frac{9-2}{3-1}\\ =\frac{7}{2}\end{array}$

The point slope form for the equation of line is given by,

  $y-y_1=m\left(x-x_1\right)$

Substitute $1$ for $x_1$ , and $2$ for $y_1$ , and $\frac{7}{2}$ for $m$ in the above formula to find equation of the line.

  $\begin{array}{c}y-2=\frac{7}{2}\left(x-1\right)\\ y=\frac{7}{2}x-\frac{3}{2}\\ f\left(x\right)=\frac{7}{2}x-\frac{3}{2}\end{array}$

Check:

Substitute $5$ for $x$ in the equation $f\left(x\right)=\frac{7}{2}x-\frac{3}{2}$ .

  $\begin{array}{c}f\left(x\right)=\frac{7}{2}\left(5\right)-\frac{3}{2}\\ =\frac{35}{2}-\frac{3}{2}\\ =16\end{array}$

Compare the equation $f\left(x\right)=\frac{7}{2}x-\frac{3}{2}$ with $y=mx+b$ .

  $\begin{array}{c}m=\frac{7}{2}\\ b=-\frac{3}{2}\end{array}$

Therefore, the value of $m$ and $b$ are $\frac{7}{2}$ , and $-\frac{3}{2}$ respectively.