~~~ 20 points ~~~

country carpets charges $22 per square yard for carpeting, and an additional installation for $100. city carpets charges $25 per square yard for the same carpeting, and an additional installation fee of $70.

a. write and solve an equation to find the number of square yards of carpeting for which the total cost charged by the two companies will be the same.

b. mr. shu wants to hire one of the two carpet companies to install carpeting in his basement. is he more likely to hire country carpets or city carpets? explain your reasoning.

A) Lets write our equations.

22x + 100 = 25x + 70

Now lets solve step-by-step.

Apply the inverse operation            22x + 100 = 25x + 70
                                                       -22x             -22x

25x - 22x = 3x

3x + 70 = 100
       -70     - 70               We subtract 70 from both sides because (inverse                                                                                                                  operation)

3x = 30

3x/3 = 30/3
x = 30/3
x= 10

B) We need to find which one is the most cheap.

Lets pick an integer to replace x.

22x + 100 = 25x + 70

Lets choose '2'

22(2) + 100 = 25(2) + 70
44 + 100
 50 + 70

144
120

The first one : Country Carpets charges $144 , while at the same rate, City carpets charges 120.

So, obviously, City Carpets is cheaper than that other one, so he would most likely choose that one.  

a.
Let x be the number of square yards of carpeting for which the total cost charged by the two companies will be the same.

We have an expression for the total cost of Country Carpets:
($22/ 1 yd^2)* (x yd^2) + $100
= $22*x+ $100
or 22x+ 100

We also have another expression for the total cost of City Carpets:
($25/ 1 yd^2)* (x yd^2)+ $70
= $25*x+ $70
or 25x+ 70

Now set the two expressions to be equal:
22x+ 100= 25x+ 70
⇒ 100- 70= 25x -22x
⇒ 30= 3x
⇒ 30/3= x
⇒ 10= x

The final answer for a is 10 square yards of carpeting~

b.
Let's assume that Mr. Shu wants to install carpeting for 1 square yard.

Country Carpets: $22*1+ $100= $122
City Carpets: $25*1+ $70= $95

Therefore, Mr. Shu will be likely to choose City Carpets if he wants to install carpeting for 1 square yard.

Now let's try x=11 (square yard):
Country Carpets= $22*11+ $100= $342
City Carpets= $25*11+ $70= $345

Therefore, Mr. Shu will be likely to choose Country Carpets if he wants to install carpeting for 11 square yard.

If you paid really close attention, you would notice that the price of City Carpets will be cheaper than the other when x is >0 and <10. At x=10, the price is equal. When x is >10, the price of Country Carpets will be cheaper than the other one.

So it depends:
If x were >0 and <10, Mr. Shu would be likely to choose City Carpets.
If x were 10, Mr. Shu could choose either Country Carpets or City Carpets.
If x were > 10, Mr. Shu would be likely to choose Country Carpets.

Hope this helps~