Systems of equations with substitution: coins

As a birthday gift, Zoey 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 read money $2.00, 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 n equal the number of nickels. Maybe I'll write "let" here. Let's let q be equal to the number of quarters. So how many total coins do we have? We'', 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 $0.25 per quarter, or 0.25 of $1. So let me write 0.25 times the number of quarters. For example, if I had 4 quarters and no nickels, I'd have 4 times $0.25 which is $1. And no money due to nickels. So it's however may nickels times $0.05 plus however many quarters times $0.25. That's the total amount of money I have. And her piggy bank tells me that is $2.00. That is equal to $2.00. So we have two equations with two unknowns. We can solve for n and q. And let's do it by substitution. So the easiest thing that we could do here, let's solve for q over here. So if n plus q is equal to 16, we could subtract n from both sides of this equation. So if n plus q is equal to 16, if we subtract n 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 quarters, we can replace it with 16 minus n. So let's do that. So the second constraint when we make the substitution becomes 0.05n plus 0.25. Instead of q, I'm going to write 16 minus n. That's what the first constraint tells us. q must be 16 minus n. That is going to be equal to $2.00. We're solving this system by substitution. Now let's see if I can simplify this. So we have 0.05n plus-- let's distribute the 0.25 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 0.25n. And that is going to be equal to $2.00. Let me scroll down a little bit. I'll scroll down a little bit. See we have 0.05n minus 0.25n. So if I have 0.05 minus 0.25, let me combine these terms. So if I have 0.05 of something, and I'm going to subtract from that 0.25 of that something, that'll give me negative 0.20 of that something. If I combine these two terms, I get negative 0.20 or negative 0.2n. And then of course, I have the plus 4. Plus 4 is equal to $2.00, or we could even just write 2 there. Now, we can isolate the n on 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 is negative 0.20n is equal to 2 minus 4 is negative 2. And then we could divide both sides by negative 0.2. Or I could write negative 0.20, the same thing. We're not going to go too deep into the significance in all that. We're assuming that we have infinite precision on everything. So negative 2 divided by negative 0.2, these guys cancel out, and we are left with n is equal to-- the negatives cancel out. 2 divided by 0.2 is just going to be 10. n is equal to 10. And then we know that q is equal to 16 minus n from the first constraint. q is equal to 16 minus n, which is 10, which is going to be 6. So Zoey put in 10 nickels. I want to do that in a different color. She put in 10 nickels and 6 quarters in the bank. And we can verify it. So clearly she has 16 coins. So that part makes sense. 10 nickels, 6 quarters, that's 16 coins. That makes sense. And we could also verify that it's the right amount of money. 10 nickels are going to be $0.50, 10 times $0.05 each. So it's going to be $0.50. And then 6 quarters is going to be $1.50. So it's going to be $1.50. So the total amount of money she has is $0.50 plus $1.50 which is $2.00. So it all works out.