<PRE><B>Question 110196</B>
You need to create expressions that describe both the number of bills and the value of those bills.
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We want to start by defining the variables and since the question is asking how many of each type of bill he has, that gives us a good basis to do this.  So:
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Let o = the number of ones
Let f = the number of fives, and
Let t = the number of tens.
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One piece of information we have is that he has a total of 95 bills and that he only has ones, fives, and tens.  So we can write:
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Eq 1) {{{o+f+t=95}}}
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Now, the value of the one dollar bills that he has is also o, because each one dollar bill has a value of one dollar.  The value of the five dollar bills is 5f, because each five has a value of five dollars, and similarly the value of his ten dollar bills is 10t.  Since we know that the total amount of money he has is $575, we can write:
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Eq 2) {{{o+5f+10t=575}}}
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So far, so good.  Now, twice the number of five dollar bills is 2f, and five more than that is 2f + 5, and we are told that is equal to the number of ones plus the number of tens, o + t.  Now we can write:
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{{{o+t=2f+5}}}, but we need to rearrange this so that this equation is in the same form as the other two by subtracting 2f from both sides, thus:
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Eq 3) {{{o-2f+t=5}}}
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So now we have three linear equations in three variables.  Let&#39;s see if there is a solution.
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Step 1, subtract Eq 3) from Eq 1), term by term:
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{{{0o+3f+0t=90}}}, and simplify
{{{3f=90}}}
{{{f=30}}}
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Now we know that Daniel has 30 fives and we can substitute this information into Eq 1) and Eq 2), thus:
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Eq 4) {{{o+30+t=95}}}, which simplifies to {{{o+t=65}}},
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Eq 5) {{{o+5(30)+10t=575}}}, which simplifies to {{{o+10t=425}}}
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Solving Eq 4) for o, we get {{{o=65-t}}}, and then substituting into Eq 5) we get:
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{{{(65-t)+10t=425}}}, then solve for t:
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{{{65+9t=425}}}
{{{9t=360}}}
{{{t=40}}}
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And now we know that Daniel has 40 tens.  Substitute that into Eq 5) to get:
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{{{o+10(40)=425}}}
{{{o=25}}}
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Giving us the last piece of the puzzle: Daniel has 25 ones.
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Now we have to check the answer.
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25 ones plus 30 fives plus 40 tens is, in fact, 95 bills.
$25 in ones, plus $5 times 30 = $150 in fives, plus $10 times 40 = $400 in tens is 25 + 150 + 400 which does, in fact, equal $575.  So the answer checks, and perhaps you are a little less confused.
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