Question 1091336
<br>Imagine one customer making the first purchase of 7 pounds of jelly beans and 9 pounds of almonds for $35; and imagine a second customer making the second purchase of 5 pounds of jelly beans and 3 pounds of almonds for $19.<br>
Now another customer comes along and makes a purchase that is exactly 3 times what the second customer bought -- 15 pounds of jelly beans and 9 pounds of almonds, for $57.  That purchase has the same number of pounds of almonds as the first customer; so the difference in what they paid must be because of the different numbers of pounds of jelly beans.<br>
The third customer paid $22 more than the first, and the difference between their purchases was an extra 8 pounds of jelly beans.  So the cost for one pound of jelly beans must be $22 divided by 8, or $2.75.<br>
Knowing the price for each pound of jelly beans, we can use the purchase of any one of the three customers to determine the cost for each pound of almonds.<br>
The first customer bought 7 pounds of jelly beans, at $2.75 per pound, for a total of $19.25.  The cost of her 9 pounds of almonds was therefore $35 - $19.25 = $15.75.  And so the cost of each pound of almonds was $15.75 divided by 9, or $1.75.<br>
So the cost of one pound of jelly beans is $2.75, and the cost for one pound of almonds is $1.75.<br>
The preceding discussion was just a way of explaining in real world terms an algebraic solution, which looks like the following.<br>
Let x = cost of one pound of jelly beans
Let y = cost of one pound of almonds
Then
(1) {{{7x + 9y = 35}}}
(2) {{{5x + 3y = 19}}}
(3) {{{15x + 9y = 57}}}  (multiplying the second equation by 3)
(4) {{{8x = 22}}}  (subtracting equation (1) from equation (3))
(5) {{{x = 2.75}}}
(6) {{{7(2.75) + 9y = 35}}}  (substituting (5) into (1))
(7) {{{19.25 + 9y = 35}}}
(8) {{{9y = 15.75}}}
(9) {{{y = 1.75}}}<br>
Those formal algebraic steps exactly followed the informal solution shown previously.