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Linear or Non-Linear?

Of the six relationships below which are perfectly linear and which are non-linear?

i) Quantity Profit
14 $7,000
0 $0
19 $9,000
11 $5,500

ii) Quantity Metal Used
7 17.5 oz
0 0 oz
9 23.5 oz
3 7.6 oz

iii) Quantity Fxed Expeneses
3 oz $4,200
0 oz $4,200
5 oz $4,200
1 oz $4,200

iv) Quantity Labor Used
0 0 hrs
7 52.5 hrs
10 75 hrs
2 15 hrs

v) Shoes Belts
0 19
5 4
6 1
1 16

vi) Quantity Metal Used
10 33.0 oz
15 49.5 oz
7 23.1 oz
9 29.7 oz

vii) A=B^2 + 2

viii) y=x

Solution Preview

The linear relationship shows constant slope, i.e., Y = a + bX
dY/dX = b
(where "d" means "changes in")

i) Quantity Profit
14 $7,000
0 $0
19 $9,000
11 $5,500

When Q changes from 0 to 14, dQ = 14, and dProfit = 7000
Then dProfit/dQ = 7000/14 = 500
However, from 14 to 19, dQ = 19-14 = 5, but dProfit = 9000-7000 = 2000
Then dProfit/dQ = 2000/5 = 400
So the slope is not constant, the relationship is NOT linear.

ii) Quantity Metal Used
7 17.5 oz
0 0 oz
9 23.5 oz
3 7.6 oz
When Q changes from 0 to 3, dQ = 3, and dY = 7.6
Then dY/dQ = 7.6/3 = ...

Solution Summary

Clear calculations and reasoning to decipher the linearness or non-linearness of 6 given relationships.

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