Chapter 1.3, Problem 33E

To determine

Tofind:The missing values in the table given below for $y=2x-3$ .

$x$	$y$	Change $\left(\Delta y\right)$
$1$	$?$	$?$$?$$?$
$2$	$?$
$3$	$?$
$4$	$?$

Answer to Problem 33E

The complete table for $y=2x-3$ is shown below:

$x$	$y$	Change $\left(\Delta y\right)$
$1$	$-1$	$2$$2$$2$
$2$	$1$
$3$	$3$
$4$	$5$

Explanation of Solution

Given information:

The equation is $y=2x-3$ and the table is:

$x$	$y$	Change $\left(\Delta y\right)$
$1$	$?$	$?$$?$$?$
$2$	$?$
$3$	$?$
$4$	$?$

Calculation:

Substitute 1 for $x$ in the given equation.

  $\begin{array}{c}y=2x-3\\ y=2\times 1-3\\ y=2-3\\ y=-1\end{array}$

Substitute 2 for $x$ in the given equation.

  $\begin{array}{c}y=2x-3\\ y=2\times 2-3\\ y=4-3\\ y=1\end{array}$

Substitute 3 for $x$ in the given equation.

  $\begin{array}{c}y=2x-3\\ y=2\times 3-3\\ y=6-3\\ y=3\end{array}$

Substitute $4$ for $x$ in the given equation.

  $\begin{array}{c}y=2x-3\\ y=2\times 4-3\\ y=8-3\\ y=5\end{array}$

The change $\left(\Delta y\right)$ for the values $-1$ and 1 is:

  $\begin{array}{c}\Delta y_1=y_2-y_1\\ =1-\left(-1\right)\\ =2\end{array}$

The change $\left(\Delta y\right)$ for the values 1 and 3 is:

  $\begin{array}{c}\Delta y_2=y_3-y_2\\ =3-1\\ =2\end{array}$

The change $\left(\Delta y\right)$ for the values 3 and 5 is:

  $\begin{array}{c}\Delta y_3=y_4-y_3\\ =5-3\\ =2\end{array}$

So, the values of $\Delta y$ are $2$ , $2$ and $2$ .

Therefore, the complete table is given below:

$x$	$y$	Change $\left(\Delta y\right)$
$1$	$-1$	$2$$2$$2$
$2$	$1$
$3$	$3$
$4$	$5$