MATH QUESTIONS, please help..... with answers and solutions...?

Question

1. in a clothing store, all sweaters are sold at one fixed price and all shirts are sold at another fixed price. If one sweater and 3 shirts cost $42, while three sweaters and two shirts cost $56, find the price of one sweater and one shirt.

2. The perimeter of a tennis court is 228 feet. After a round of tennis, a players coach estimates that the athlete has run a total of 690 feet, which is equivalent to 7 times the court's length plus 4 times its width. What are the dimensions of a standard tennis court?

3. Traveling downstream w/current, it takes a motorboat 2 hours to cover distance of 48 miles. When the motorboat returns upstream against the current, it takes 3 hours to cover the same distance. Find the speed of the motorboatin still water and the speed of the current.

Answer 1

#1

Say SH is the shirt cost and SW is the sweater cost:
SW + 3SH = 42
3SW+2SH = 56

Solve the first for SW:
SW = 42-3SH

Plug that into the second:
3(42-3SH) +2SH = 56

Simplify:
126 - 9SH +2SH = 56
70 = 7SH
SH = 10

Plug this back into the SW formula:
SW = 42 - 3SH = 42 - 3*10 = 12

Shirts cost $10, sweaters cost $12.

#2
Call L the length and W the width:
P = 2L + 2W = 228
7L + 4W = 690

Solve the first for W:
2W = 228 - 2L
W = 114 - L

Plug that into the second:
7L + 4(166 - L) = 690
7L + 456 - 4L = 690
3L = 234
L = 78

Plug this into the formula for W
W = 114 - L = 114 - 78 = 36

The court is 78ft x 36ft.

#3
Call S the speed of the boat and C the speed of the current.

With the current, the total speed is:
48miles / 2 hours = 24 mph

So S + C = 24

Against the current, the total speed is:
48 miles / 3 hours = 16

So S - C = 16

We can "add" equations by adding the left and right hand sides, so that we get:

S+C + S-C = 40
2S = 40
S = 20

Looking at:
S + C = 24

We know C = 4.

The current is 4mph, and the boat goes 20 mph in still water.

Answer 2

These are known as simultaneous equations in 2 unknowns.

1) Let the fixed price of a sweater be X.
Let the fixed price of a shirt be Y.
Then:
a. X + 3Y = 42
b. 3X + 2Y = 56
Multiply a by 3:
c. 3X + 9Y = 126
Subtract b from c:
7Y = 70, 0r Y = 10
Price of shirts is $10
Substitute in a:
X + 30 = 42
X = 42 - 30, X = 12
Price of sweaters is $12
Proof:
Substitute in b:
36 + 20 = 56
left side = right side.

2) Let the length of the tennis court be L feet.
Let the width of the tennis court be W feet.
Perimeter: 2L + 2W = 228
Estimate: 7L + 4W = 690
Multiply Perimeter by 2:
Double: 4L + 4W = 456
Subtract from Estimate:
3L = 234, L = 78 feet
Substitute in Perimeter:
156 + 2W = 228
2W = 228 - 156 = 72
W = 36 feet
Proof:
Substitute in Estimate:
7x78 + 4x36
546 + 144 = 690
left = right side

You can try the third one.

Answer 3

1. x+3y=42
3x+2y=56
multiply first equation by -3 to get
-3x-9y=-126 combine with equation 2
3x+2y=56

-7y=-70 y=10 shirts are $10 so sweaters are $12 by substitution.

2. 37 by 77 similar set up as to number 1
2l+2w=228
14l +8w=690 multiply 1st by -4 and combine.

3.
with current 24 mph
against 16 mph

still water is 20 mph with the current being 4 mph.

Answer 4

1. s= sweater t=shirt

1s + 3t= 42
3s + 2t= 56

You have a systems of equation.

-3 ( 1s +3t = 42)
3s+2t = 56


-3s-9t= -126
3s + 2t =56

-7t = -70
t=10

To find s, substitute t.
1s + 3t = 42
1s + 3(10) =42
1s+ 30 =42
s= 12

Sweaters are $12, Shirts are $10.



2. length =x width= y

2x+ 2y = 228
7x + 4y= 690

Systems of equations.

-2 (2x +2y =228)
7x +4y = 690 =

-4x-4y = -456
7x +4y =690
3x =234
x= 78

length = 78
2x+ 2y = 228
2(78) +2y = 228
156 + 2y = 228
y= 36,
width = 36