Question 558772: In one week, a music store sold 7 violins for a total of $1600. two different tyoes were sold.One cost $200 and the other cost $300. How many of each type were sold?
PLZ HELP THIS IS DUE 2MORROW ON 1-18-12
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Let x be the unknown number of $200 violins sold and let y be the unknown number of $300 violins sold.
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Ask yourself, what two things about the situation does the problem tell you? These two things gives you the clues that lead to the solution.
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The first thing is that a total of 7 violins was sold. That tells you that x plus y must equal 7. So write that as one equation:
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x + y = 7
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The next thing you are told is that the money received by the store for selling the violins was a total of $1600.
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Think about this. For each of the cheaper violins, the store received $200. x of these violins were sold. So if you multiply x times 200 dollars you get the amount of money the store collected for these violins. Similarly y of the more expensive violins were sold. Since each of these cost 300 dollars, if you multiply the number of these violins (y) times 300 you get the amount of money the store collected for them. The total must be 1600 dollars. So now we can write another equation:
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200x + 300y = 1600
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We now have two independent equations and two unknowns. Since the number of independent equations is the same as the number of unknowns, we should be able to solve this problem.
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One way to solve this set of two equations is by substitution. We can look at the first equation and solve it for x in terms of y. Start with:
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x + y = 7
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Subtract y from both sides:
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x + y - y = 7 - y
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On the left side the +y and the -y cancel each other out and we are left with:
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x = 7 - y
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Now we can go to our second equation and substitute 7 - y for x because we know that they are equal. So start with:
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200x + 300y = 1600
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and replace x with the quantity 7 - y and this equation becomes:
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200(7 - y) + 300y = 1600
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Now we have one equation that has only one unknown. So we can solve for that one unknown.
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Begin by multiplying the 200 times the quantity 7 - y. To do that distributed multiplication, you multiply the 200 times both the 7 and the -y and you 1400 and -200y. So the equation now becomes:
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1400 - 200y + 300y = 1600
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Get rid of the constant 1400 on the left side by subtracting 1400 from both sides. On the left side the 1400 disappears and on the right side the 1600 minus the 1400 results in 200. So the equation has now been reduced to:
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-200y + 300y = 200
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On the left side the -200y combines with the +300y to give 100y. This further reduces the equation to:
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100y = 200
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Now we can solve for y by simply dividing both sides of the equation by 100. When we do that we get:
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y = 200/100
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and that becomes:
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y = 2
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And y represented the number of $300 violins sold. Since we now know that, and since we know that there were 7 violins sold, we can say that there were 5 of the $200 violins sold (7 minus 2 = 5).
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Let's check that answer. If 5 of the $200 violins sold, the total amount received from those sales was 5 times $200 and this equals $1000. And 2 of the $300 violins sold for a total of 2 times $300 which equals $600. Adding those two totals ($1000 and $600) results in $1600, just as the problem indicated that it should. Therefore, our answer is correct. We can now say that we know the store sold two of the $300 violins and five of the $200 violins.
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I hope this gives you some insight into how you can solve problems such as this one. Make sure that you understand all the steps and why we did them. Good luck ...
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