Vintage US Postcards

How would I solve these two word problems?

A shopper pays $20.75 for four pounds of a nut mixture consisting of almonds and pecans. If almonds cost $3.50 per pound and pecans cost $6.00 per pound, how much of each were purchased?

Solve as a system of equations where x = weight of Almonds and y = weight of Pecans
The second equation is weight multiplied by price to give the total price of the mixed nuts.

x + y = 4
3.5x + 6y = 20.75
-------------------------

-6(x + y = 4) .....multiply by (-6) so that you can add the 2 equations isolating x

-6x - 6y = -24
3.5x + 6y = 20.75
-----------------------
-2.5x + 0y = -3.25
-2.5x/-25 = -3.25/-2.5 ... divide both sides by -2.5 to isolate x
x = 1.3

through substitution now find y
1.3 + y = 4
y = 4 - 1.3
y = 2.7

check your answer by inserting both values into the 2nd equation

3.5(1.3) + 6(2.7) = 20.75
4.55 + 16.20 = 20.75 check ... 1.3lbs Almonds 2.7lbs of Pecans

=============

2nd problem is the same type as the first

x = # of postcards y = # of letters

x + y = 56
.27x + .42y = 21.72
----------------------------

100(0.27x + 0.42y = 21.72) .... do this to eliminate the fractions.

x + y = 56
27x + 42y = 2172
---------------------------

-27( x + y = 56) .... multiply the equation by (-27) so that you can add the 2 equations isolating y to find the number of letters

-27x - 27y = -1512
27x + 42y = 2172
------------------------------
0x + 15y = 660

15y/15 = 660/15 ...divide both sides by 15 to isolate y
y = 44

through substitution now find x

x + 44 = 56
x = 56 - 44
x = 12

check your answer by inserting both values into the 2nd equation

.27(12) + .42(44) = 21.72
3.24 + 18.48 = 21.72 ......... check ... 12 postcards .. 44 letters

edited for clarity

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