Question 596839
Hi there--
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[I] Define your variables
Let's have h be the number of half-dollars in the coin bank.
Let's have d be the number of dimes in the coin bank.

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The value of the half-dollars is $.50 times the number of half-dollars there are in the bank. One half-dollar is 0.50, two half-dollars are 2*(0.50)=$1.00, three are 3*(0.50)=$1.50. The algebraic expression is 0.50*h. 
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Using similar reasoning, the value of the dimes in the bank is 0.10*d since one dime is $0.10.
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Set-up your equations:
First make an equation for the total number of coins. There are 24 coins in all, so
[the number of half-dollars] + [the number of dimes] = 24
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In algebra, we write,
{{{h+d=24}}}
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Now we will make an equation for the total value of the coins in the bank, $3.60.
[the value of the half-dollars]  +  [the value of the dimes]  = $3.60
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In algebra, we write,
{{{0.50h+0.10*d=3.60}}} 
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[III] Solve the system of equations
I will use the substitution method. Write the first equation in terms of d.
{{{h+d=24}}}
{{{d=24-h}}}
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Substitute 24-h for d in the second equation.
{{{0.50h+0.10d=3.60}}}
{{{0.50h+0.10(24-h)=3.60}}}

Now clear the parentheses.
{{{0.50h+2.40-0.10h=3.60}}}
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Combine like terms.
{{{0.40h+2.40=3.60}}}
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Subtract 2.40 from both sides of the equation to isolate the h-term and the constant term on opposite sides.
{{{0.40h=1.20}}}
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Divide both sides of the equation by 0.40 to isolate h.
{{{h=3}}}
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The equation h=3 means that there are 3 half-dollars. Since there are 24 coins all together, there must be 21 dimes.
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We need to make sure these coins have a total value of $3.60. Three half-dollars is $1.50, and twenty-one dimes is $2.10. $1.50+2.10=$3.60. BINGO!
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That's it. Feel free to email me via gmail if any part of this explanation is unclear.
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Ms.Figgy
math.in.the.vortex@gmail.com