Solving a tax rate or interest rate problem using a system of linear equations

Question

A theater group made appearances in two cities. The hotel charge before tax in the second city was $1500 higher than in the first. The tax in the first city was 10%, and the tax in the second city was 4%. The total hotel tax paid for the two cities was $480. 

How much was the hotel charge in each city before tax?

First city:

Second city:

Answer 1

So if you let x = hotel cost city1; and y = hotel cost city2 then:

1. (hotel tax total) 0.1X + .04Y = 480
2. (relationship between city taxes) X + 1500 = Y
3. You can now use the substitution method to solve for X (then plug that answer into either equation to solve for Y
4. 0.1(X) + .04(X + 1500) = 480 distribute and group like terms
5. .1X + .04X + 60 = 480
6. .14X = 420
7. X = $3000
8. 3000 + 1500 = Y; Y = $4,500

Does this help? Do you see how to set each of the linear equations up? (you set up by "topic" to solve).

Answer 2

Hope this helps...

Use any TWO variables for each hotel in each city; I will use x & y

Ex. y = hotel charge in city 1 with tax rate of 10%, x = hotel charge in city 2 with tax rate of 4%

You need TWO equations relating x and y with the information given...

Knowing that difference between the two hotel charges is 1500, set your first equation;

x = y + 1500 OR x - y = 1500

Your 2nd equation will be based on the tax information given. The sum of the tax charges = $380

x(0.04) + y(0.10) = 380

Time to solve: x - y = 1500

0.04x + 0.10y = 480

Working with the TOP equation, modify the equation by factoring by a factor so when you add the equations together you eliminate ONE of the variables to solve for the other. Here I will use -0.04 to cancel out the x term.

-0.04(x - y) = 1500(-0.04) = -0.04x + 0.04y = -60

Add the equations...the x term cancels out

-0.04x + 0.04y = -60

0.04x + 0.10y = 480

0.14y = 420

y = 420/0.14

y = $3,000 = hotel charge in city 1 BEFORE taxes

then

x = y + 1500 = $4,500 = hotel charge in city 2 BEFORE taxes

CHECK by calculating the tax charges:

Hotel in city 1 = 0.10(y) = $300

Hotel in city 2 = 0.04(x) = $180