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# Systems of equations with substitution: coins

Video transcript

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.