Linear Equations and Applications

For problems 1-12, solve the equation .

(1) 3(2 − x) = 2x − 1

(3) 0.9t = 0.4 + 0.1t

(4) x(1 + 2x) = (2x − 1)(x − 2)

(6) x^2 = 9x

(7) 4z^3 − 8z^2 = 0

where a, b, and c are constants and c ≠ 0, a ≠ 0.

(13) Leigh is paid time and a half for hours worked in excess of 40 hours
double time for hours worked on Sunday. If Leigh had gross weekly
wages of $342 for working 50 hours, 4 of which were on Sunday, what
was the regular hourly rate?

(14) In a basketball game, the Bulls scored a total of 103 points and made
three times as many field goals (2 points each) as free throws (1 point
each). They also made eleven 3 point baskets. How many field goals
did they have?

(15) Nancy jogs and walks to campus each day. She averages 4 kilometers
per hour walking and 8 kilometers per hour jogging. The distance from
home to the campus is 6 kilometers and she makes the trip in 1 hour.
How far does she jog on each trip?

(16) A recent retiree, Kate requires $6,000 per year in extra income. She has
$50,000 to invest and can invest in noninsured B-rated bonds paying
15% per year or in a government insured Certificate of Deposit (CD)
paying 7% per year. How much money should be invested in each to
realize exactly $6,000 in interest per year?

(17) A bank loaned out $12,000, part of it at the rate of 8% per year and
the rest at the rate of 18% per year. If the interest received totaled
$1,000, how much was loaned at 8%? What was the average rate of
return?

(18) For a certain experiment, a student requires 100 cubic centimeters of a
solution that is 8% hydrochloric acid. The storeroom has only solutions
that are 15% hydrochloric acid and 5% hydrochloric acid. How many
cubic centimeters of each available solution should be mixed to get 100
cubic centimeters of 8% hydrochloric acid?

(19) A coffee manufacturer wants to market a new blend of coffee that will
cost $3.90 per pound by mixing two coffees that sell for $2.75 and $5.00
per pound, respectively. The total amount of this new mixture will be
100 pounds worth. What amounts of each coffee should be blended to
obtain the desired mixture?

(20) How many ounces of pure water should be added to 20 ounces of a 40%
solution of muriatic acid to obtain a 30% solution of muriatic acid?

(21) The manager of the Coral Theater wants to know whether the majority
of its patrons are adults or children. During a week in July, 5200 tickets
were sold and the receipts totaled $20,335. The adult admission is
$4.75, and the children’s admission is $2.50. How many adult patrons
were there?

(22) An automobile radiator contains 16 litres of antifreeze and water. This
mixture is 30% antifreeze. How much of this mixture should be drained
and replaced with pure antifreeze so that the final mixture will 50%
antifreeze? Inherent in this problem, you must assume that you will
fill the radiator back up.

(23) How much water must be evaporated from 64 ounces of a 2% salt
solution to make a 10% salt solution?

(24) A wool suit, discounted by 30% for a clearance sale, has a price tag of
$399. What was the suit’s original price?

(25) Matthew by himself can paint four rooms in 10 hours. If Letitia helps
him, they can do the same job in 6 hours. How long would it take
Letitia by herself to paint the four rooms?

(26) Trent can deliver newspapers in 30 minutes. It takes Lois 20 minutes to
do the same route. How long will it take them to deliver the newspapers
together?

(27) A bathroom tub will fill in 15 minutes with both fauces open and the
stopper in place. With both faucets closed and the stopper removed ,
the tub will empty in 20 minutes. How long will it take for the tub to
fill if both faucets are open and the stopper is removed ?

(29) A five horsepower pump can empty a pool in 5 hours. A smaller,
2 horsepower pump empties the same pool in 8 hours. The pumps
are used together to begin emptying this pool. After two hour , the 2
horsepower pump breaks down. How long will it take the larger pump
to empty the pool?

(30) A motorboat heads upstream on a river that has a current of 3 miles
per hour. The trip upstream takes 5 hours, while the return trip takes
2.5 hours. What is the speed of the motorboat? (Assume that the
motorboat maintains a constant speed relative to the water.)

(31) A tight end can run the 100 yard dash in 12 seconds. A defensive back
can do it in 10 seconds. The tight end catches a pass at his own 20 yard
line with the defensive back at the 15 yard line. If no other players are
nearby, at what yard line will the defensive back catch up to the tight
end?

(32) Tanya, who is a long distance runner, runs at an average velocity of 8
miles per hour. Two hours after Tanya leaves your house, you leave in
your Honda and follow the same route. If your average velocity is 40
miles per hour, how long will it be before you catch up to Tanya? How
far will each of you be from your house?