Lydia wants to buy picnic tables, benches, and permanently installed grills. On one page she notices that 5 picnic tables, 10 benches, and 3 grills sell for $1330. On another page she sees that 20 picnic tables, 30 benches, and 20 grills sell for $5550. Finally, on a third page she sees that 2 picnic tables and 4 benches sell for $430. Help Lydia find the individual prices for the three items. Define three variables, write a system of equations, and solve to find the cost of each item. Justify your answer by showing how you solved the problem, and check the solution.
I will give you a kickstart.
Represent each of the items by a symbol, say:
T=picnic tables
B=bench
G=Grill
On the first page: " 5 picnic tables, 10 benches, and 3 grills sell for $1330. "
So 5T+10B+3G = 1330 .....(1)
Continue this way and you should end up with three equations in the three unknowns T, B and G.
Solve the system for Lydia.
To solve this problem, let's define three variables:
Let the cost of one picnic table be 'x'
Let the cost of one bench be 'y'
Let the cost of one grill be 'z'
Using these variables, we can write a system of equations based on the information given:
Equation 1: 5x + 10y + 3z = 1330 (from the first page)
Equation 2: 20x + 30y + 20z = 5550 (from the second page)
Equation 3: 2x + 4y = 430 (from the third page)
To solve the system of equations, we can use the method of substitution or elimination. Let's use the method of substitution:
From Equation 3, we can isolate 'x' in terms of 'y':
2x = 430 - 4y
x = (430 - 4y)/2
x = 215 - 2y
Now, substitute the value of 'x' in Equations 1 and 2 with (215 - 2y):
Equation 1: 5(215 - 2y) + 10y + 3z = 1330
1075 - 10y + 10y + 3z = 1330
1075 + 3z = 1330
3z = 1330 - 1075
3z = 255
Equation 2: 20(215 - 2y) + 30y + 20z = 5550
4300 - 40y + 30y + 20z = 5550
4300 - 10y + 20z = 5550
-10y + 20z = 5550 - 4300
-10y + 20z = 1250
We have now obtained two equations in terms of 'z':
Equation 4: 3z = 255
Equation 5: -10y + 20z = 1250
Solving Equation 4 for 'z', we find z = 85.
Now substitute the value of 'z' in Equation 5:
-10y + 20(85) = 1250
-10y + 1700 = 1250
-10y = 1250 - 1700
-10y = -450
y = -450/-10
y = 45
Finally, substitute the values of 'y' and 'z' into Equation 3 to find 'x':
2x + 4(45) = 430
2x + 180 = 430
2x = 430 - 180
2x = 250
x = 250/2
x = 125
Therefore, the cost of one picnic table (x) is $125, the cost of one bench (y) is $45, and the cost of one grill (z) is $85.
To check our solution, we can substitute these values back into any of the original equations. Let's use Equation 1:
5x + 10y + 3z = 1330
5(125) + 10(45) + 3(85) = 1330
625 + 450 + 255 = 1330
1330 = 1330
The left-hand side is equal to the right-hand side, confirming that our solution is correct.
Therefore, the individual prices for the picnic table, bench, and grill are $125, $45, and $85, respectively.