SOLUTION: I'm having some trouble solving these problems please help me. Use algebraic techniques to solve rate problems and percent problems: 1.) Five tables and eight chairs cost $11

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Question 151442: I'm having some trouble solving these problems please help me.
Use algebraic techniques to solve rate problems and percent problems:
1.) Five tables and eight chairs cost $115; three tables and five chairs cost $70. Determine the cost of each table and each chair?
2.)A merchant sold his entire stock of shirts and ties for $1000, the shirts being priced at 3 for $10 and the ties at $2 each. If he sold 1/2 of the shirts and 2/3 of the ties he would have collected $600. How many of each kind did he sell?
3.)Determine the speed of a motorboat in still water and the speed of the river current, if it takes 3 hr to travel a distance of 45 mi upstream and 2 hr to travel 50 mi downstream?

Please and thank you
Answer by ankor@dixie-net.com(22740) (Show Source):
You can put this solution on YOUR website!
1.) Five tables and eight chairs cost $115; three tables and five chairs cost $70. Determine the cost of each table and each chair?
:
Let x = cost of tables; let y = cost of chairs
Write an equation for each statement:
"Five tables and eight chairs cost $115;"
5x + 8y = 115
"three tables and five chairs cost $70."
3x + 5y = 70
Use elimination here, Multiply the 1st equation by 3, the 2nd equation by 5
15x + 25y = 350
15x + 24y = 345
-------------------subtraction eliminates x, find y
0x + 1y = 5
y = $5 is cost of one chair
:
Find x using the 1st equation:
5x + 8(5) = 115
5x + 40 = 115
5x = 115 - 40
5x = 75
x =
x = $15 is cost of one table
;
Check solutions in 2nd equation
3(15) + 5(5) =
45 + 25 = 70
:
:
2.)A merchant sold his entire stock of shirts and ties for $1000, the shirts being priced at 3 for $10 and the ties at $2 each. If he sold 1/2 of the shirts and 2/3 of the ties he would have collected $600. How many of each kind did he sell?
:
Let x = no. of shirts sold
let y = no. of ties
:
Write an equation for each statement:
"A merchant sold his entire stock of shirts and ties for $1000, the shirts being priced at 3 for $10 and the ties at $2 each"
10() + 2y = 1000
+ 2y = 1000
multiply equation by 3 to get rid of the denominator
10x + 6y = 3000; (1st equation)
:
"If he had sold only 1/2 of the shirts and 2/3 of the ties he would have collected $600. How many of each kind did he sell?"
* + *2y = 600
+ = 600
Reduce fraction
+ = 600
Multiply equation by 3 to get rid of the denominator
5x + 4y = 1800
:
multiply above equation by 2 and subtract the 1st equation:
10x + 8y = 3600
10x + 6y = 3000
-----------------subtraction eliminates x
0x + 2y = 600
y = 300 ties sold
:
Find x, use eq: 10x + 6y = 3000
10x + 6(300) = 3000
10x + 1800 = 3000
10x = 3000 - 1800
10x = 3000 - 1800
10x = 1200
x =
x = 120 shirts sold
:
Check solution in
5x + 4y = 1800
5(120) + 4(300) =
600 + 1200 = 1800
:
:
3.)Determine the speed of a motorboat in still water and the speed of the river current, if it takes 3 hr to travel a distance of 45 mi upstream and 2 hr to travel 50 mi downstream?
:
Let x = speed of boat in still water
Let y = speed of the current
then
(x+y) = speed with the current
and
(x-y) = speed against the current
:
Write a distance equation for each trip: Dist = time * speed
;
2(x + y) = 50
3(x - y) = 45
;
Simplify divide the 1st equation by 2 and the 2nd equation by 3; results:
x + y = 25
x - y = 15
------------addition eliminates y, find x
2x = 40
x =
x = 20 mph speed of boat in still water
:
Find y using; x + y = 25
20 + y = 25
y = 25 - 20
y = 5 mph speed of the river
:
:
Check solutions in the 2nd original equation
3(x - y) = 45
3(20-5) = 45
3(15) = 45; confirms our solution