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# Systems of equations with substitution: coins

CCSS.Math:

## Video transcript

as a birthday gift Zoe gave her niece an electronic piggy bank that displays the total amount of money in the bank as well as the total number of coins after depositing some number of nickels and quarters only so we only have nickels and quarters the display red money \$2 number of coins 16 how many nickels and quarters did Zoey put in the bank so let's define some variables here let's let let's let n equal the number of nickels and let's all right let here let's let Q be equal to the number of quarters the number of quarters so how many total coins do we have well it's going to be the number of nickels plus the number of quarters so we have the nickels plus the quarters need to be equal to well it tells us we have 16 total coins so if we add up the total number of nickels plus the number of quarters we have 16 coins so that's one equation right there and then how much total money do we have well however many nickels we have we can multiply that times 0.05 and that'll tell us how much money we have in nickels so 0.05 times the nickels plus the amount of money we have in quarters well that'll just be 25 cents per quarter or 0.25 of a dollar so let me write 0.25 times the number of quarters for example if I had four quarters and no nickels I'd have four times 25 cents which is one dollar and no money do two nickels so however many nickels times five cents plus however many quarters times 25 cents that's the total amount of money I have and her piggy bank tells me that that is two dollars that is equal to two dollars so we have two equations with two unknowns we can solve for N and Q and let's do it by substitution so let's see the easiest thing that we could do here let's solve let's solve for Q over here so if n plus Q is equal to 16 we can subtract n from both sides of this equation so if n plus Q is equal to 16 if we subtract and from both sides we get Q is equal to 16 minus n so all I did is I rewrote this first constraint right over there so since this first constraint is telling us that Q the number of quarters must be 16 minus the number of nickels in the second constraint every place that we see a Q every place we see a quarters we can replace it with 16 minus n so let's do that so the second constraint when we make the substitution becomes 0.05 n plus 0.25 instead of Q I'm going to write 16 minus n 16 minus n that's what the first constraint tells q must be 16 minus n that is going to be equal to \$2 we're solving the system by substitution now let's see if I can simplify this so 0.05 n plus let's distribute the point two five times the 16 and the 0.25 times the negative n 0.25 times 16 that's the same thing as 1/4 times 16 that's just going to be 4 and then 0.25 times negative n is minus 0.25 N and that is going to be equal to two dollars let me scroll down a little bit actually I could continue I'll scroll down a little bit see we have 0.05 n minus 0.25 and so if I have to point O 5 minus 0.25 so let me combine these terms so if I have point O 5 of something and I'm going to subtract from that point to 5 of that something that'll give me negative zero point to 0 of that something if I combine these two terms I get negative point to 0 or negative 0.2 and then of course I have the plus 4 plus 4 is equal to \$2 or we could even just write 2 there now we can isolate the end of the left-hand side by subtracting 4 from both sides so let's subtract 4 from both sides and we are left with on the left-hand side negative I could just write that as negative zero point to 0 n is equal to 2 minus 4 is negative 2 is equal to negative 2 then we can divide both sides by negative point to divide both sides by negative zero point two or I could write negative point zero negative zero point two zero the same thing we're not going to go too deep into the significant digits in all of that we're assuming that we have infinite precision on everything so negative two divided by negative zero point two these guys cancel out and we are left with n is equal to the negatives cancel out two divided by point two is just going to be ten n is equal to ten and then we know that Q is equal to sixteen minus n from the first constraint q is equal to sixteen minus n which is ten which is going to be six so Zoey put in ten nickels she put in ten nickels I want to do that in a different color she put in ten nickels and six quarters and six quarters in the bank and we can verify it so clearly she has 16 coins so that part makes sense ten nickels six quarters that 16 coins that makes sense and we could also verify this the right amount of money 10 nickels 10 nickels are going to be 50 Cent's ten times ten times five cents each so it's going to be 50 cents and then six quarters six quarters is going to be a dollar fifty so it's going to be a dollar fifty so the total amount of money she has is 50 cents plus a dollar fifty which is two dollars so it all works out