Systems of Equations Part I

Systems of Equations (Part I)

Answer the question and solve the problems below. Make sure you show all your work so you can get partial credit even if you get the final answer wrong.

Best Rentals charges a daily fee plus a mileage fee for renting its cars. Barney was charged $123.00 for 3 days and 300 miles, while Mary was charged $216.00 for 5 days and 600 miles. What does Best Rental charge per day, for mileage?

Let x = daily fee and y = mileage fee. The system of equations would then be as follows: Barney (3x + 300y = $123.00), Mary (5x + 600y = $216.00).

We take Barney's equation and isolate for x:

300y = 123 (3x + 300y)/3 = 123/3 x + 100y = 41 x = 41-100y

Using substitution for Mary's equation, we get

5(41-100y) + 600y = 216-205 -- 500y + 600y = 216-205 + 100y = 216 100y = 11

Therefore, y = $0.11 per mileage

To find x, we plug into Barney's equation:

x = 41 -- 100y x = 41 -- 100(0.11) x = 41 -- 11 = $30.00 per day.

In conclusion, Best Rental charges $30.00 for a daily fee and $0.11 per mile.

2. There were 44,000 people at a ball game in Los Angeles. The day's receipts were $369,000. How many people paid $13.00 for reserved seats and how many paid $6.00 for general admission?

Let x = people with reserved seats and y = people with general admission. This means that x + y = 44,000 people, and that 13x + 6y = $369,000. The system of equations is seen as follows:

Equation 1: x + y = 44,000

Equation 2: 13x + 6y = 369,000

Isolating for x, we get:

x + y = 44,000 x = 44,000 -- y

Substituting for Equation 2:

13x + 6y = 369,000 13(44,000 -- y) + 6y = 369,000 572,000 -- 13y + 6y = 369,000 572,000 -- 7y = 369,000 -7y = -203,000 y = 29,000

Plugging in for x:

x = 44,000 -- y x = 44,000 -- 29,000 = 15,000

In conclusion, 15,000 people paid for $13 reserved seating, and 29,000 people paid for $6 general admission.

3. Solve the system of equations x + 3y = 32

-3x + 2y = 3

Isolating for x for the first equation, we get x + 3y = 32 x = 32 -- 3y

Substituting for the second equation, we get

-3x + 2y = 3 -3(32-3y) + 2y = 3 -96 + 9y + 2y = 3 11y = 99 y = 9

Plugging in to the x equation, we get x = 32 -- 3y x = 32 -- 3(9) x = 32 -- 27 x = 5

Therefore, x = 5, y = 9.

4. Solve the system of equations x + y = 5

x - y = -9

Using a different method (addition), one solves the following system of equations:

x + y = 5

x -- y = -9

(x + y) + (x -- y) = 5 + (-9) 2x = -4 x = -2

Plugging in for y, we get x + y = 5 y = 5 -- x y = 5 -- (-2) y = 7

Therefore, x = -2, y = 7