# Smart Nails vs. Polish'd

### Who offers the better deal?

Bianca and Amaya want to get their nails done for the school dance coming up. They're on a strict budget so they have to find a good deal. They looked at 2 of their favourite nails salons in the G.T.A, "Smart Nails" and "Polish'd". At “Smart Nails” you can get a manicure for \$30 plus \$2 for a design per nail. At “Polish’d” you can get a manicure for \$25 plus \$3 for a design per nail. Which is a better deal if Amaya wants a manicure + design on 2 nails and if Bianca wants a manicure + designs on all 10 of her nails?

## Variables

Let Y represent cost, X represent the # of nails with designs.

## Equations

Smart Nails: y=2x+30

Polish'd: y+3x+25

The table lists and compares the prices at the two salons.
The graph and table above compares the prices at the two salons. By looking at it you can see the point that the lines intersect, which is at 5 nails with designs. This is where both salons offer the same price for the same number of nails. The data shows that before 5 nails with designs Polish'd has a better deal and after 5, Smart Nails has a better deal.

## How to find the Point of Intersection

First you find the x value. You do this by writing an algebraic equation where the equation for Smart Nails equals the equation for Polish'd, and solving it.

2x+30 = 3x+25

30-25 = 3x-2x

5 = x

Now that you've found the value of x. Replace it with x in both equations to find the value of y. They should be the same.

Smart Nails:

y=2x+30

y=2(5)+30

y=10+30

y=40

Polish'd:

y=3x+25

y=3(5)+25

y=15+25

y=40

so x= 5 and y=40 and the point of intersection is (5,40)

## Summary

Based on the prices above it would be cheaper for Amaya to get her nails done at Polish'd and for Bianca to get her nails done at Smart Nails.