Chapter 4.2, Problem 12PPE

To determine

Whether the relationship shown in the table is linear.

Represent the relationship using words, an equation, and a graph.

Answer to Problem 12PPE

Yes, relationship is linear.

Explanation of Solution

Given information:

The following table is given.

x	y
0	$\$52.07$
1	$\$53.36$
2	$\$54.65$
3	$\$55.94$

Calculation:

x	y
0 $(x_1)$	$\$52.07$ $(y_1)$
1 $(x_2)$	$\$53.36$ $(y_2)$
2 $(x_3)$	$\$54.65$ $(y_3)$
3 $(x_4)$	$\$55.94$ $(y_4)$

The relationship is said to be linear if one variable $y$ changes by consistent amount as the other variable $x$ increases.

Now check whether the relationship is linear or not.

  $\frac{y_2-y_1}{x_2-x_1}=\frac{53.36-52.07}{1-0}=1.29$

  $\frac{y_3-y_2}{x_3-x_2}=\frac{54.65-53.36}{2-1}=1.29$

  $\frac{y_4-y_3}{x_4-x_3}=\frac{55.94-54.65}{3-2}=1.29$

As $\frac{y_2-y_1}{x_2-x_1}=\frac{y_3-y_2}{x_3-x_2}=\frac{y_4-y_3}{x_4-x_3}$ , relationship is linear.

Now find the equation.

Take $(x_1,y_1)=\left(0,52.07\right),\left(x_2,y_2\right)=\left(1,53.36\right)$

Equation is given by $\left(y-y_1\right)=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$

  $\begin{array}{l}\left(y-52.07\right)=\frac{53.36-52.07}{1-0}\left(x-0\right)\\ \Rightarrow y-52.07=1.29x\\ \Rightarrow y=52.07+1.29x\end{array}$

In words, the relationship can be defined as follows:

Total bill is $\$52.07$ more than $1.29$ times of number of soup cans.

Now draw the graph.

First find points that can be used to plot the graph.

x	y
0	$\$52.07$
1	$\$53.36$
2	$\$54.65$
3	$\$55.94$

So, plot the points $(\mathrm{0.52.07}),(1,53.36),(2,54.65),(3,55.94)$ and join them