Linear systems w/ 3 variables

Question

Larry has a total of 24 nickels, dimes, and quarters whose total value is $3. The number of nickels equals the sum of the number of dimes and quarters.

a. Write 3 equations modeling the total number of coins, the value of the coined, and the relationship between the number of nickels and other coins.

b. Solve the system of equations to determine how many of each coin Larry has.

Answer 1

a. So for the three equations, use x as your number of nickels, y as your number of dimes, and z as your number of quarters.

1. .05x + .10y + .25z = 3.00
2. x + y + z = 24
3. y + z = x or -x + y + z = 0

b. Now let's pick a variable that would be easy to eliminate. x looks easy. If you add the bottom two equations together, you get 2y + 2z = 24. If you multiply the top equation by -20, you can then add the top two equations together. -1y + -4z = -36. Now with these two new equations, you can solve for z by eliminating y.

2y + 2z = 24

2(-1y + -4z = -36).

-6z = -48

z = 8

Plug z back into one of your equations now, 2y + 2(8) = 24, and solve for y.

y = 4

Now plug y and z into one of your original equations and solve for x.

x + y + z = 24

x + 4 + 8 = 24

x = 12

So z=8, y=4, and x=12. Larry has 8 nickels, 4 dimes, and 12 quarters.