Question

colleen has 12 coins in her pocket. the mix of quarters, nickles and dimes add up to two dollars, and she has three times as many quarters as nickles

How many of each coin does Colleen have in her pocket
(solve this problem using Gaussian elimination)

Answer 1

By solving a system of equations we conclude that Collen has 6 quarters, 4 dimes, and 2 nickels.

How many of each coin does Colleen have in her pocket?

First, let's define the variables:

• x = number of quarters.
• y = number of nickels.
• y = number of dimes.

There are 12 coins, so:

x + y + z = 12

She has 2 dollars in total, so:

x*0.25 + y*0.05 + z*0.10 = 2

And she has 3 times as many quarters as nickels.

x = 3y

Then we can write the system of equations (or matrix, depending on how you like to see it):

x + y + z = 12

x*0.25 + y*0.05 + z*0.10 = 2

x - 3y = 0

To use the elimination method we just need to add/subtract these equations to remove variables.

If we subtract the third equation to the first one:

(x + y + z) - (x - 3y) = 12 - 0

4y + z = 12

Now we have two equations:

x*0.25 + y*0.05 + z*0.10 = 2

4y + z = 12

If we multiply the first one by 4:

x + y*0.20 + z*0.40 = 8

Subtracting the third eq:

x + y*0.20 + z*0.40 - (x - 3y) = 8 - 0

y*3.20 + z*0.40 = 8

Now the remaining equations are:

4y + z = 12

y*3.20 + z*0.40 = 8

On the top one, we can rewrite: z = 12 - 4y

Replacing that in the other equation:

y*3.20 + (12 - 4y)*0.40 = 8

Now we eliminated the variables x and z, so we can solve this for y:

y*1.60 + 4.8 = 8

y*1.60 = 3.2

y = 3.2/1.6 = 2

Now that we know the value of y, the values of x and z are:

x = 3y = 3*2 = 6

z  = 12 - 4y = 12 - 4*2 = 4

Then we conclude that Collen has 6 quarters, 4 dimes, and 2 nickels.

If you want to learn more about systems of equations:

https://brainly.com/question/13729904

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