Holiday Shopping!

Which day is the best deal for you?

It's That Time of Year....

During the holiday season, millions of people around the world go shopping to share Christmas joy with their loved ones. In addition to being festive, the holiday season can also be quite expensive for many buyers so these shoppers decide to shop on Boxing Day for Christmas presents. Every year, more and more peopled decide to do their Christmas shopping on boxing day to get the best deals and save money. Here is how the data looks like....

Variables

Let x represent the year.

Let y represent the number of products.

Gifts Bought on Boxing Day Data Table

Equation of Line

Slope and y intercept form:

y = mx + b

y = 30x + 200


Standard form:

Ax + Bx + C = 0

30x + y + 200 = 0

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Gifts Bought on Christmas Data Table

Equation of Line

Slope and y intercept form:

y = mx + b

y = -50x + 900


Standard form:

Ax + Bx + C = 0

50x + y - 900 = 0

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Linear System Comparison

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Question

In what year will there be the same amount of products sold on each of the lists?

Solution - Solving for a Point of Intersection

To find in what year the 2 days will have the same number of products sold, we must solve the linear system, which is essentially finding a point of intersection.


1. We can solve by graphing

Using the equations given for each line, we can graph each line using the y intercept and rise over run for slope.

Christmas Sales

y = -50x + 900

- (0,900) is the y intercept so plot it

- Then use rise over run to graph the slope which is -50

- Graph the line

Boxing Day Sales

y = 30x + 200

- Repeat same steps


Then, using the 2 graphed lines, find the point of intersection. On the graph, the point of intersection is in the year of 2008 with



2. You can also solve these linear systems by substitution (algebraically)


y = -50x + 900

y = 30x + 200


Sub in one equation into the other and solve for x

-50x + 900 = 30x + 200

900 - 200 = 30x + 50x

700 = 80x

700/80 = x

x = 8.75


Now sub x back into any equation and solve for y

y = 30(8.75) + 200

y = 262.5


Therefore, the point of intersection is (8.75 , 262.5)



Hence, in the year of 2008, there were 262 (262.5 to be exact) products sold on each day.

Why Use a Linear System?

A linear system helps track the rate at which something increases or decreases at. In this case, the linear systems help us see how many products are sold each year on either Boxing Day or Christmas. Also, we can use the progression of the linear system to predict how many products will be sold in the future. This data can help companies maximize their profits depending on their prices and which day they choose to sale it.